Salvatore Spina
email: salvatore.spina@unict.it...

Filemaker, Gephi, Princes Ignazio Transkribus

Michele Maria Paternò

THE BISCARI ARCHIVE. A CASE STUDY OF THE APPLICATION OF THE TRAN-SKRIBUS TOOL

The 'Paternò-Castello Principi di Biscari' Archive, preserved at the State Archives of Cataniaamongst one of the most crucial family archivesis, in the light of a digital historical methodology, the best computable historical heritage for demonstrating the applicability of applying an HTR tool, such as Transkribus, to digitised historical documents.

INTRODUCTION

The archive of the Paternò-Castello family, princes of Biscari, kept at the State Archives of Catania, represents a valuable set of computable historical data. Moreover, its heterogeneity may support the hypothesis that it is possible to transform the archival heritage of History into Big Data. This means that all kinds of historical information can be digitised. Therefore, a proper digitisation process would help create a historical documentary apparatus capable of assisting the historian in his efforts to reconstruct past events. Due to its characteristics, precisely because it is the cultural, social, and political expression of a dynamic community, the Biscari Archive stores documents with unique peculiarities that a historian cannot disregard. Applying sophisticated computer tools is the only way to bring out all the data from the sources. Indeed, human physiological limitations do not allow for the analysis of large amounts of data and information. Fortunately, the development of artificial intelligence, deep learning tools and neural networks can compensate for this inherent difficulty in human nature. However, some clarifications are in order. First and foremost, historical documentation of the Modern Era is composed mainly of manuscripts, representing an obstacle, as they are, to the possibility of using computer tools. Archival documentation is mainly composed of manuscripts, which is a significant obstacle to the possibility of using computer tools for a study concerning an event that took place in the Modern Age. Secondly, a lot has already been said about digitisation and digitalisation. However, the two terms reproduce a conception in most scholars: "digitisation" is like "taking a picture". According to "analogue" historians, this is one of the assumptions on which to build the Digital History Statute'. Most historians believe that the digital methodology consists of scanning or photographing some archival document and uploading it to a database. Indeed, one of the initial stages of any historical study is the retrieval and organisation of the. However, in a digital dimension that digitises objects, places and people, a photo cannot be read by Artificial Intelligence in the same way the historian reads the text of the image. For a computer, a photo is a set of data, such as the colour codee.g. #0000 is 'black', #654321 is 'brown', #FF0000 is 'red', and so on -, the definition, the number of pixels, the place where it was takenif the camera is equipped with GPSand other similar information1 .

Nevertheless, if we consider the text of any manuscript, no computer can figure out all the information it carries because it is inaccessible even to the Turing machine. If this were the case, the computer could only reproduce words without relation to reality because language is a code structured on non-explicit elements, inconsistency between word and action and other aspects that computers cannot process. Furthermore, even if a computer can analyse a digital text, it cannot read and recognise a handwritten word, not even a single consonant or vowel. Still, this assumption brings us to some key points: is photographing the right way to digitise an archival heritage? Can a computer read handwritten text? How could digitisation and computer tools change the craft of historians? A study on historical sources of the Biscari Archive -Catania, Italy2 can help us solve the dilemma and demonstrate the feasibility of using HTR technology at various stages of historical research. Within this archive are 2,000 folders with hundreds of thousands of sheets documenting legal disputes, political decisions, and business and personal letters. The 'Correspondence' section consists of some 42,493 papers, in which, in folder 1642, we find 591 sheets, constituting 366 letters and a manuscript by Emile Rousseau. Although a 'quantitative' approach would have been more appropriate to choose the initially computable documentary materialsuch as an "income and expenditure register"a folder displaying non-numerical information was opted for. This is because the creation of an innovative model of historical research, as demonstrated by Ruth and Sebastian Ahnert and Kim Albrecht (https//tudornetworks.net), requires the development of technological systems capable of analysing heterogeneous data, such as those chosen in our case. The documents acquisition was performed with the Nikon D610 equipped with the AF-S Nikkor 24-120mm f/4G ED VR lens, and the photos taken were collected in a database created with Claris Filemaker 19, a software that was chosen since a website is not planned to be built at this stage of the project. After the metadata3 insertion phase, the photos were converted to PDF format to reconstruct the 366 letters, and 53 of these were uploaded to the Transkribus server to obtain the automatic transcription.

TRANSKRIBUS & BISCARI ARCHIVE

Due to its Java technology, Transkribus4 (https://readcoop.eu/transkribus/?sc=Transkribus) allows the creation of workflows based on "deep neural networks"5 , which can be trained to recognize a particular handwriting. One of the options required to get the best result is that the texts are written by the same hand. For this reason, 53 letters were chosen, 28 letters from Michele Maria Paternò -Prior of Messinato Princess Anna Maria Morso Bonannowife of Ignazio Paternò, fifth prince of Biscari -, and 25 letters from Marquis Giovanni Fogliani Sforza D'Aragona, Viceroy of Sicilia, sent to different addresses. The 53 letters were selected after analysing the network of people (fig. 1) who corresponded with the Prince of Biscari or other members of the house of Paternò-Castello. The database configuration allowed us to index each person and place the 367 records into a relational structure. Thanks to the geo-coordinates and the application of 'The Vistorian', 'Palladio', and 'Gephi' tools, it was possible to (1) place the various senders in territories (e.g. Sicily, Italy, Europefig. 2); (2) determine the different relations between the data (e.g. sender-consignee).

Figure 2. The network of the place.

However, a crucial question persists: how does a historian identify a group of letters written by the same hand? One of the fundamental aspects of Digital History is its ability to narrate by "visualising". Through an editorial design product, i.e., it is possible to display information which allows a scholar to be guided towards a comprehensive view of the object of study. Moreover, among the most helpful and widely used approaches, Historical Network Analysis (HNA) decisively meets historians' need to make sense of information. The HNA postulates that men build relationships with fellow men and the surrounding spaceeven in the absence of other subjects. Wherever there is a man, there is a relation. Where there is a 'node' (subject), there is an 'arc' (edge; the relationship). This schema functions as a human paradigm (i.e. "the network framework shapes the way we interpret the world around us") that becomes the foundation of the Network Analysis approach: everything can be reconstituted visually into an open schemathat can be continuously filled inthat provides different information depending on the object of research. Thanks to Gephi, it was possible to identify an appropriate number of letters written by the same sender (fig. 3)thus by the same handensuring the uniformity required by the Transkribus tool. Within the Biscari Archive, precisely at folder 1642, section 'Correspondence', can be found 28 letters written by Michele Maria Paternò, Prior of Messina. He was a religious man very close to the Biscari family, especially to Princess Anna, the addressee of this correspondence. While the other 25 letters chosen for the test were written by Marquis Fogliani. Both groups of letters (a total of 112 sheets) were sorted and merged into two separate pdf files, which were subsequently uploaded to the Transkribus server. The small size of the corpus certainly facilitated rapid analysis and automatic transcription, even though the tool's purpose is to allow the transcription of larger archival material. After all, Transkribus is meant to meet one of the benchmarks of historical research methodology in the digital environment: automatically transcribing many handwritten texts that historians would never be able to record during their research.

Here, the number of letters is sufficient to understand whether Traskribus is able to transcribe them correctly and, above all, whether the HTR model for the Italian language has been adequately traineddespite the inevitable reluctance of the Italian scientific community to use state-of-the-art IT tools. Whilst the international scientific community aspires to build an adequately digitised historical heritage, the Italian's is failing to break out of the constraints of the legal protection of archival heritage, leading to a delay in digitising historical heritage. In addition, Italy lacks a scientific mindset to ensure the dissemination of its national archival heritage, unlike, for example, Finland, where, thanks to the National Archives project, the Central Government wanted to restore to its community the archival heritage related to the Second World War, the judicial registers and property inventories of the Finnish nobility. To do this, it involved students and scholars in training a model to obtain transcriptions with minimal margins of error. Detailed and in-depth training will enhance artificial intelligence, enabling Transkribus to automatically transcribe the entire human heritage in all languages and eras of the past. A further core feature of Transkribus is that digital transcription projects of archival heritages can be open to the entire community, who thus participate in the training phases, thus realising the grand goal of Public History.

OUTCOME

The automatic transcription of the two sets of letters took place without any training activity because the primary intent was to understand the accuracy achieved by the public Transkribus AI model. For transcription, we applied the "Italian Administrative Hands 1550-1700" model created by Jake Dyble, Antonio Iodice, Sara Mansutti and Rachel Midura, whose CER (Character Error Rate) is 9.15 per cent. This rate represents an excellent result for a public model in the automatic transcription of manuscripts kept in Italian archives. The latter, in fact, hold the most heterogeneous historical heritage in Europe, not only because of the different writing styles but also because of the presence of different languages and institutionse.g., Latin in the Vatican, Spanish in the Kingdom of Sicily and the Duchy of Milan, Italian in the Grand Duchy of Tuscany. The model is applied and transcribed when the two PDF files are uploaded to the Transkribus server. In the case of Marquis Fogliani's letters, the error rate is 7% (e.g., in the first letter, 6 out of 82 words are not transcribed correctly), and the interval remains constant for almost all sheets. The error interval is much higher for the 28 letters written by Michele Maria Paternò. In fact, it exceeds 10% due to the presence of incorrectly written Italian words and the presence of words unknown to the Transkribus model. Also, in the second letter, it can be seen that the graphemes "V.S." and "S.M," although present in the segmentation of the text image, are not transcribed. This condition results in a higher error rate because the tool not only did not transcribe even incorrectly but did not allow any correction by the user. The latter cannot identify the word in the window intended for digital transcription and its correction. Given the numerous transcription errors, it was decided to train a model based on the "Italian Administrative" model, starting with the 24-page "ground truth" (pages 1-5, 7-9, 11-18, 20-24, 26, 27, 35). Once completing the training and forming the model -"Michele Maria Paternò_Archivio Bis-cari_Model" -, the CER of the Training Set dropped to 0.68% and that of the Validation Set to 9.42% (fig. 4). From the comparison (e.g., page 38 -see Figure 5) between the "ground truth "(manual) transcription and the automatic one, on the model just mentioned, the CER stands at 9.88%, yet that of the validation set returns a relatively high percentage (35.57%), which is indicative of training that needs to be carried out on a more substantial ground truth (50 pages, at least) to provide a quantitatively and qualitatively accurate transcription. However, as can be seen (Figure 7), the application of the "Michele Maria" model resulted in a transcript whose CER is 8.07%/26.29 and with minor errors, which do not compromise the scholars' inferring but make the computational analysis unreliable.

TRANSKRIBUS E FILEMAKER

(1) A research project in History starts with the organisation of primary sources. (2) "Analogue" historians have to transcribe the sources, analyse them in detail and try to seize all the meanings through their intuitive and intellectual faculties alone. (3) "Digital" historians use digital archives and databases, in which he arranges sources and the information they have drawn from them. (4) Each stage can take advantage of computer technology, allowing historians to process more information and coordinate many more sources.

The study conducted at Biscari Archive documents aims to show how these technologies can dialogue with each other and enable historians to reconstruct past events thanks to more articulate and detailed information. Transkribus allows transcriptions to be exported in PDF files, which can be analysed, processed, and organised in databases. This enables searches through an interface and a relational system highlighting the novel information. Thus, Transkribus makes it one of the essential tools for building the Big Data of History. Moreover, it responds to Gardin's assumption that the creation of databases and ontologies cannot be separated from the correct 'encoding and formalisation' phase, which enables the scholar to codify historical texts into a computable version. To this aim, the 53 transcriptions from the Biscari archive were entered into a database created with Filemaker 19 (fig. 6), a software that enables the creation of a relational structure between the data within it. For its part, Filemaker, with the update to version 19.5, has also implemented a new JSON function, which allows for the differentiation of numbers and text and the use of system libraries, OCR and HTR images inserted in a container fielda possibility that not only reflects the need to deal with the digitisation of photographic sources but also makes the databases ready for the creation of Machine-Readable sources. Transkribus aims to demonstrate the need for upgrading historians' skills in a digital environment, focusing on the most sensitive and essential aspects of source analysis: organisation and transcription.

ACHIEVEMENT

The term 'digitisation' has pervasively entered the vocabulary of our society. This concept, however, is very often misused. Firstly, three concepts are lumped together (digitisation, digitalisation, and digital transformation). Secondly, for "analogue" humanists, this term is synonymous with "facsimile", i.e. an image to be observed on a monitor, without considering that the digitisation process is quite different from the simple act of photographing/scanning an archival document.

Every document is an aggregate network compound of pieces of information, and only its proper digitalisation will allow us to create digital texts that can enrich our knowledge of past events, thanks to tools such as Network Analysis, which allows the intuitive processes of the historian to reach a level previously unattainable. Networks of people, places and events become, in fact, the necessary background for a historical ontology, but the necessary step remains, irrefutably, the automation of the transcription of manuscript sources on which historical research is based. The information stored in a photographed text escapes the computer. To the latter, images are just meaningless sets of pixels. It is the semantic context of a source, the meaning of words, that makes a document instrumental to reconstructing a historical event. Thus, the Biscari Archive is an essential asset to reconstructing the History of Catania and of many events that marked the History of Sicily. However, every piece of information contained therein is yet to be identified and analysed. For this reason, our trial is intended as a call to look at the archival heritage in the light of Digital History and ICT paradigms. Historical research requires an essential investment in digitising archival sources, considering that the digital world requires a specific configuration (through Transkribus and other tools) that goes beyond simple photographic acquisition.

Figure 1 .

 1 Figure 1. Sender and consignee.

Figure 3 .

 3 Figure 3. Correspondence between Michele Maria Paternò and Anna Maria Morso Bonanno.

Figure 4 .

 4 Figure 4. Correspondence between Michele Maria Paternò and Anna Maria Morso Bonanno.

Figure 5 .

 5 Figure 5. Comparison set.

Figure 6 .

 6 Figure 6. The Database, Transkribus and Filemaker.

  REFERENCEH. Aghdam, E. Heravi, Guide to Convolutional Neural Networks. A Practical Application to Traffic-Sign Detection and Classification, 2017 G. Amato, F. Falchi, L. Vadicamo, Visual Recognition of Ancient Inscriptions Using Convolutional Neural Network and Fisher Vector, in «Journal on Computing Cultural Heritage», fasc. 9, 4 2016, pp. 1-24 M. Boillet, C. Kermorvant, T. Paquet, Multiple Document Datasets Pre-training Improves Text Line Detection With Deep Neural Networks, in «2020 25th International Conference on Pattern Recognition (ICPR)», 2021, pp. 2134-2141 G. Calabrese, L'archivio della famiglia Paternò Castello principi di Biscari: inventario, Os. n.!, Catania 2003 M. C. Calabrese, Il Gran Priore dell'Ordine di Malta e il suo servo. Un episodio di schiavitù mediterranea nel XVIII secolo, in «Nuova rivista storica», fasc. 100, 3 2016, pp. 907-976 F. D'Avenia, Le commende gerosolimitane nella Sicilia moderna: un modello di gestione decentrata, 2000, pp. 35-86 P. Kahle, S. Colutto, G. Hackl, G. Mühlberger, Transkribus. A Service Platform for Transcription, Recognition and Retrieval of Historical Documents, 2017 14th IAPR International Conference on Document Analysis and Recognition (ICDAR), 2017vol. 04, , pp. 19-24 N. Milioni, Automatic Transcription of Historical Documents. Transkribus as a Tool for Libraries, Archives and Scholars, 2020 S. Spina, Digital History. Metodologie informatiche per la ricerca storica, Edizioni Scientifiche Italiane, Napoli 2022

S. Spina, Digital History. Metodologie informatiche per la ricerca storica, Edizioni Scientifiche Italiane, Napoli

G. Calabrese, L'archivio della famiglia Paternò Castello principi di Biscari: inventario, Os. n.!, Catania 2003.

Sender, consignee, date and place of issue, place of destination, named persons and places, regesto and keywords.

P.Kahle et al., Transkribus. A Service Platform for Transcription, Recognition and Retrieval of Historical Documents, 2017 14th IAPR International Conference on Document Analysis and Recognition (ICDAR), 2017, v. 04, pp. 19-24; N. Milioni, Automatic Transcription of Historical Documents. Transkribus as a Tool for Libraries, Archives and Scholars, 2020.

H. Aghdam, E. Heravi, Guide to Convolutional Neural Networks. A Practical Application to Traffic-Sign Detection and Classification, 2017; G. Amato et al., Visual Recognition of Ancient Inscriptions Using Convolutional Neural Network and Fisher Vector, in «Journal on Computing Cultural Heritage», 9, 4 2016, pp. 1-24; M. Boillet et al., Multiple Document Datasets Pretraining Improves Text Line Detection With Deep Neural Networks, in «2020 25th International Conference on Pattern Recognition (ICPR)», 2021, pp. 2134-2141.


Amaury Freslon

Rubén Martos

TORSION AND K-THEORY FOR SOME FREE WREATH PRODUCTS

Keywords: 2010 Mathematics Subject Classication. 20G42, 46L80, 19K35 Quantum groups, K-theory, Baum-Connes conjecture

HAL is

Introduction

This paper is concerned with K-theory for C*-algebras associated to discrete quantum groups. Since the fundamental work of M. Pimsner and D.-V. Voiculescu [START_REF] Pimsner | K-groups of reduced crossed products by free groups[END_REF] proving that C * r (F n ) C * r (F m ) if n = m as a corollary of the computation of their K 1 -groups, the computation of K-theory groups of reduced C*-algebra has been an important subject. A very general method to carry these computations by geometric means was developed under the name of Baum-Connes conjecture (see [START_REF] Baum | Classifying space for proper actions and K-theory of group C*-algebras[END_REF]). The idea is to build an assembly map between the K-theory on the right-hand side and the K-homology of a toplogical space associated to the group on the left-hand side. The conjecture is that this map is an isomorphism. Even though the strongest forms of the conjecture are known to fail [START_REF] Higson | Counterexamples to the BaumConnes conjecture[END_REF], it is still a powerful method for a large class of groups, for instance those satisfying the Haagerup property.

If we turn to discrete quantum groups, the very denition of the assembly map and the statement of a proper Baum-Connes conjecture is unclear. D. Goswami and A. Kuku gave in [START_REF] Goswami | A complete formulation of the Baum-Connes conjecture for the action of discrete quantum groups[END_REF] a non-commutative version of the classifying space for proper actions of a discrete quantum group together with an assembly map. However, there is no method available to prove that the assembly map is bijective or to compute the K-homology of the left-hand side. Forgetting the geometric aspects of the problem and focusing on the homological algebra, R. Meyer and R. Nest gave in [START_REF] Meyer | The Baum-Connes conjecture via localisation of categories[END_REF] a version of the Baum-Connes conjecture in terms of the triangulated structure of the equivariant Kasparov category. Using this, they were able to prove it for duals of compact Lie groups.

The rst application of this method to genuine quantum groups is due to C. Voigt in [START_REF] Voigt | The Baum-Connes conjecture for free orthogonal quantum groups[END_REF], where he proved the Baum-Connes conjecture for free orthogonal quantum groups and computed their K-theory. Later, he adapted with R. Vergnioux ideas of G. Kasparov an G. Skandalis [START_REF] Kasparov | Groups acting on buildings, operator K-theory, and Novikov's conjecture[END_REF] to prove in [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF] the Baum-Connes conjecture for arbitrary free products of free unitary and orthogonal quantum groups and compute their K-theory. All these cases have an important feature : they are torsion-free in a K-theoretic sense. In the presence of torsion, things become more complicated and the only results so far are also due to C. Voigt [START_REF]On the structure of quantum automorphism groups[END_REF] for quantum automorphism groups of nite-dimensional C*-algebras.

Our goal in the present paper is to investigate the Baum-Connes conjecture for the free wreath product of an arbitrary compact quantum group G by the quantum permutation group S + N , as dened by J. Bichon in [START_REF] Bichon | Free wreath product by the quantum permutation group[END_REF]. This study naturally splits into two parts :

• The rst one is the computation of the torsion of the corresponding discrete quantum groups. This is a dicult problem in general, but Y. Arano and K. De Commer introduced in [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF] a powerful method to show that a quantum group is torsion-free. Free wreath products are never torsion-free, but we will see that their method can still give enough information on the torsion to yield a complete classication. Following the ideas of [START_REF]On the structure of quantum automorphism groups[END_REF], it is then easy to deduce that G * S + N has the so-called strong Baum-Connes property if G is a torsion-free compact quantum group already satisfying the strong Baum-Connes property.

• The strong Baum-Connes property means that we have concrete homological tools to compute the Ktheory. However, one still has to nd an explicit projective resolution of the trivial action, preferably of length one. We show that this can be done for several choices of G provided that we do not consider G * S + N but the monoidally equivalent quantum group G * SO q (3). Let us end this introduction with a short overview of the organization of the paper. Section 2 contains some background on compact quantum groups and the formalism of based rings. We then classify in Section 3 the torsion actions of any free wreath product G * S + N . We also investigate the behaviour of torsion actions under free complexication, enven though this is not needed in the sequel. As a consequence, we prove in Theorem 3.19 that if G is torsion-free and has the strong Baum-Connes property, then G * S + N also has the strong Baum-Connes property. Eventually, in Section 4 we compute the K-theory of G * SO q (3) for several quantum groups G.

Acknowledgments. It is a pleasure to thank Roland Vergnioux for stimulating discussions on the content of Section 4. The rst author also thanks Yuki Arano for fruitful conversations on Section 3.

Preliminaries

In this section we will recall some basic denitions and facts concerning compact quantum groups and torsion, mainly to x notations. The reader may refer for instance to [START_REF] Neshveyev | Compact quantum groups and their representation categories[END_REF] for details and proofs. A compact quantum group G is given by a C*-algebra C(G) together with a * -homomorphism ∆ : C(G) → C(G) ⊗C(G) satisfying the coassociativity condition (∆ ⊗ id) • ∆ = (id ⊗∆) • ∆ and such that both (1 ⊗ C(G))∆(C(G)) and (C(G) ⊗ 1)∆(C(G)) span a dense subspace of C(G) ⊗ C(G). Our main focus will be representations of such objects. By the results of [START_REF] Woronowicz | Compact matrix pseudogroups[END_REF], these representations are always equivalent to a direct sum of nitedimensional unitary ones, so that we will only dene the latter.

Denition 2.1. A unitary representation of dimension

n of G is a unitary element u ∈ M n (C) ⊗ C(G) such that ∆(u ij ) = n k=1 u ik ⊗ u kj .
A morphism between representations u and v of dimension n and m respectively is a linear map T : C n → C m such that (T ⊗id)u = v(T ⊗id). Two representations are said to be equivalent if there is a bijective morphism between them. Eventually a representation is said to be irreducible if its only self-intertwiners are scalar multiples of the identity.

As an example, there is always a trivial representation ε G = 1 ⊗ 1 ∈ C ⊗ C(G). The equivalence classes of irreducible nite-dimensional unitary representations of G form a set denoted by Irr(G). One way to gather all the information about the representation theory of G is to consider its representation category Rep(G) whose objects are all nite-dimensional representations and morphisms are representation morphisms, which is a rigid tensor C*-category. If F ⊂ Irr(G), one can consider the smallest full rigid tensor subcategory C F of Rep(G) containing F . By Woronowicz's Tannaka-Krein duality [START_REF]Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups[END_REF], there is then an associated C*subalgebra C(H) ⊂ C(G) such that restricting the coproduct to C(H) endows it with the structure of a compact quantum group H. Moreover, Rep(H) naturally identies with C F so that we will say that H is the quantum discrete subgroup of G generated by F .

Dening torsion (or torsion-freeness) for discrete quantum groups is not straightforward. The denition we use emerged from the developement of an analogue of the Baum-Connes conjecture for quantum groups. This conjecture concerns the equivariant KK-theory of C*-algebras acted upon by G, so that we need a dynamical characterization of torsion, which we recall from [START_REF] Meyer | Homological algebra in bivariant K-theory and other triangulated categories I[END_REF] (see also [START_REF]On the structure of quantum automorphism groups[END_REF]). Denition 2.2. A torsion action (A, α) of a compact quantum group G is given by a nite-dimensional C*-algebra A together with an action α :

A → A ⊗ C(G) which is ergodic, i.e. {a ∈ A, α(a) = a ⊗ 1} = C.1.
As an example, if γ ∈ Irr(G) and u γ ∈ B(H γ ) ⊗ C(G) is a representative, we can dene a torsion action α γ on B(H γ ) by setting α γ (T ) = (u γ ) * (T ⊗ 1)u γ . If G is the dual of a discrete group Γ, then all irreducible representations are one-dimensional and the above action is always trivial. For a general quantum group, it is trivial up to equivariant Morita equivalence. Denition 2.3. Let G be a compact quantum group. A torsion action of G is said to be trivial if it is equivariantly Morita equivalent to the trivial action. If all torsion actions are trivial, then G is said to be torsion-free. Remark 2.4. If G is the dual of a discrete group Γ, then it is torsion-free in the above sense if and only if Γ is torsion-free. This comes from the fact that nite-dimensional ergodic coactions of Γ always arise from nite subgroups of Γ, see for instance [START_REF]On the structure of quantum automorphism groups[END_REF]Prop 4.2].

Classifying torsion actions is not an easy task in general. In [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF], Y. Arano and K. De Commer developed a method to deal with this problem in a combinatorial way, through the notion of fusion ring. Since this will be fundamental in the present work, we recall the denitions in detail. The reader may nd a complete exposition of the subject in [9, Chap 3], where these objects are called Z + -rings but we will rather use the conventions and terminology of [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]. Let I be a set with a distinguished element ε and an involution i → i xing ε. A ring structure ⊗ on Z I is given by structure constants λ i 3 i 1 ,i 2 such that

i 1 ⊗ i 2 = i 3 ∈I λ i 3 i 1 ,i 2 i 3 ,
where all but nitely many terms vanish. Let us write

i 3 ⊂ i 1 ⊗ i 2 if λ i 3 i 1 ,i 2 = 0.
Denition 2.5. A I-based ring is a ring structure ⊗ on Z I with positive integer structure constants such

that • i 1 ⊗ i 2 = i 2 ⊗ i 1 , • ε ⊂ i 1 ⊗ i 2 if and only if i 1 = i 2 . A dimension function on a based ring is a unital ring homomorphism d : (Z I , ⊗) → R such that • d(i) > 0 for all i ∈ I, • d(i) = d(i).
A based ring endowed with a dimension function is called a fusion ring.

Let G be a compact quantum group, set I = Irr(G) and endow it with the involution coming from the conjugation of representations. The tensor product of representations turns Z I into a based ring and it is a fusion ring when endowed with the dimension function d of representations. This ring will be denoted by R G in the sequel and it is the only type of fusion ring we will be interested in. The main idea of [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF] is that torsion actions give rise to specic modules over the fusion ring of a compact quantum group. As before, if J is a set, then any Z I -module structure on Z J is given by structure constants λ j 2 i,j 1 such that

i ⊗ j 1 = j 2 ∈J λ j 2 i,j 1 j 2 ,
where again all but nitely many terms vanish. We will write

j 2 ⊂ i ⊗ j 1 if λ j 2 ij 1 = 0.
Denition 2.6. A J-based module is a (Z I , ⊗)-module structure on Z J with positive integer structure constants such that j 1 ⊂ i ⊗ j 2 if and only if j 2 ⊂ i ⊗ j 1 . It is said to be conite if for all j, j ∈ J, {i ∈ I, j ⊂ i ⊗ j} is nite. It is said to be connected if for any j, j ∈ J, there exists i ∈ I such that j ⊂ i ⊗ j.

A compatible dimension function on a based module is a linear map

d : (Z J , ⊗) → R such that • d(j) > 0 for all j ∈ J, • d(i ⊗ j) = d(i)d(j)
for all i ∈ I and j ∈ J. A based module endowed with a compatible dimension function is called a fusion module.

Note that if M is a conite based module over a fusion ring R with dimension function d, then we can dene a Z I -valued bilinear form by

j 1 , j 2 = i∈I λ j 2 ij 1 i.
which is equivariant in the sense that i ⊗ j 1 , j 2 = i ⊗ j 1 , j 2 . Then, for any j 0 ∈ J the map d : j → d( j, j 0 ) is a compatible dimension function on M .

Example 2.7. Let R = Z I be a based ring. The regular action turns it into a based R-module and the dimension function on the fusion ring yields a compatible dimension function on the module. It follows from the denition of a based ring that this module is conite and connected. A fusion R-module M is said to be standard if it is isomorphic to R with this fusion module structure.

Let us now explain the idea of [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]. If (A, α) is a torsion action of a compact quantum group G, then the category of G-equivariant Hilbert modules over A is a module C*-category over the representation category of G. As a consequence, its Grothendieck group is a based module over R G . The nite-dimensionality of the action implies that the module is conite, while the ergodicity yields connectedness. If the action is trivial, then the module category is exactly the representation category of G, hence the corresponding module is R G . This justies the following denitions : Denition 2.8. Let R be an I-based ring. A based module is said to be a torsion module if it is conite and connected. The based ring R is said to be torsion-free if any torsion module is standard.

According to [1, Thm 2.8], a compact quantum group G is torsion-free if R G is torsion-free. However, the converse may not hold since it is not clear that any torsion module of R G can be traced back to a torsion action of G. However, we will see that knowing the torsion modules of R G is often enough to understand the torsion actions of G. Let us conclude by a remark on the link between torsion actions and monoidal equivalence which will be crucial. If G and H are monoidally equivalent compact quantum groups in the sense of [START_REF] Bichon | Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups[END_REF], then there is a one-to-one correspondence between their actions which was explicitly described in [START_REF] De Rijdt | Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries[END_REF]. Moreover, this correspondence can be lifted to an equivalence of categories which preserves nitedimensionality and ergodicity as explained in [START_REF] Voigt | The Baum-Connes conjecture for free orthogonal quantum groups[END_REF]Sec 8]. Therefore, the equivariant Morita equivalence classes of torsion actions are also in one-to-one correspondence. This means that we only have to nd a simple monoidally equivalent model to study torsion in a given compact quantum group.

Classification of torsion actions

In this section we will investigate how torsion actions behave under several constructions. The main goal is to classify the torsion actions of free wreath products in order to be able to prove a suitable version of the Baum-Connes conjecture for them. The general strategy is as follows : we rst work at the level of fusion rings and fusion modules, where everything can be treated combinatorially. Then, we use these results to deduce the corresponding statements for genuine actions using the setting of module C*-categories and Tannaka-Krein duality for ergodic actions of compact quantum groups developed in [START_REF] Commer | Tannaka-Krein duality for compact quantum homogeneous spaces I. General theory[END_REF].

3.1. Free product. Given two compact quantum groups G 1 and G 2 , S. Wang dened in [START_REF] Wang | Free products of compact quantum groups[END_REF] their free product G 1 * G 2 (here we are slightly abusing notation since this should be understood as the dual of the free product of the dual discrete quantum groups). Since we will not need the explicit denition but only the representation theory, let us recall what irreducible representations of G 1 * G 2 look like. They can be indexed by alternating words in Irr(G 1 ) \ {ε 1 } and Irr(G 2 ) \ {ε 2 } with the following fusion rules :

wβ 1 ⊗ β 2 w = wβ 1 β 2 w
if β 1 and β 2 are not representations of the same factor and

wβ 1 ⊗ β 2 w = γ⊂β 1 ⊗β 2 γ =ε wγw ⊕ δ β 1 ,β 2 (w ⊗ w )
otherwise. It was proven in [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]Thm 1.25] that if G 1 and G 2 are torsion-free, then their free product also is. However if (A, α) is a torsion action for say G 1 , then it is also a torsion action for G 1 * G 2 through the inclusion

A ⊗ C(G 1 ) ⊂ A ⊗ C(G 1 * G 2 ).
Thus, the best that we can expect in general is that torsion actions of the free product all arise from this construction. This is what we are going to prove now. To do this, let us rst introduce some terminology. If G and H are two compact quantum groups with C(H) ⊂ C(G), then any action α of H gives rise to an action of G called the induced action and denoted by Ind G H (α). At the level of fusion modules, this corresponds to the usual construction of an induced module, namely considering the R G -module R G ⊗ R H N , which will be denoted by Ind R G R H (N ). For convenience, let us give a more explicit description in the case of free products. For two base sets I 1 and I 2 , we will denote by W (I 1 , I 2 ) the set of alternating words ending in I 2 . Lemma 3.1. Let R 1 and R 2 be two based rings with respective base sets I 1 and I 2 and let N be a based

R 1 -module with base set J. The induced module Ind R 1 * R 2 R 1 (N ) is the based module with base set {wj, j ∈ J, w ∈ W (I 1 , I 2 ) ∪ {∅}} with the obvious action of R 1 * R 2 .
Proof. This directly follows from the description of the representation theory.

Note that inducing the standard module of one of the factors yields the standard module of the free product. As a rst step, we will prove that if we induce a non-standard torsion module, then we again obtain a non-standard one. This requires a characterization of standard module through the following notion of stabilizer : if R is an I-based ring and M is a J-based R-module, we dene the stabilizer of an element j ∈ J by

Stab(j) = {i ∈ I, j ⊂ i ⊗ j}.
Note that if M is conite, then the stabilizer of any element is nite. Moreover, if the module is standard then there is at least one element with trivial stabilizer, which is the one corresponding the trivial representation of G. In fact, the converse also holds : Lemma 3.2. Let N be a fusion module of R G and assume that there is a basis element j 0 with trivial stabilizer.

Then, there is an isomorphism of based R G -modules N → R G sending j 0 to the trivial representation.

Proof. Recall from Section 2 the denition of the bilinear form j, j = β∈Irr(G) λ j βj β. By denition, if j 0 has trivial stabilizer then j 0 , j 0 = ε. Now let β ∈ Irr(G) be non-trivial and set

β ⊗ j 0 = n k=1 λ k j k .
We then get

k λ k j k , j 0 = β ⊗ j 0 , j 0 = β ⊗ j 0 , j 0 = β ⊗ ε = β.
However, j k , j 0 always contains β and λ k is a non-negative integer. The only possibility is to have k = 1. In other words, β ⊗ j 0 = j 1 is a basis element. It is now clear that sending j 0 to the trivial representation gives an isomorphism with the standard module.

This will be useful in the proofs of the main results of this section. We will also need to know how the induced module splits back over

R G 1 . Lemma 3.3. Let N be a torsion module for R G 1 . Then, M = Ind G 1 * G 2 G 1 (N ) splits as a R G 1 -module into a direct sum of N and standard modules. In particular, Ind G 1 * G 2 G 1 (N ) Ind G 1 * G 2 G 1 (N ) as R G 1 * G 2 modules if and only if N N as R G 1 -modules.
Proof. Let us consider an arbitrary R G 1 -submodule P of M and let w be a word of minimal length such that there exists a basis element j of N satisfying wj ∈ P . Assume that w = ∅ and write w = γ k . . . γ 1 . We cannot have

γ k ∈ Irr(G 1 ) since otherwise γ k-1 • • • γ 1 j ⊂ γ k ⊗ wj ∈ P , contradicting the minimality of w. Thus, γ k ∈ Irr(G 2 )
and the description of the fusion rules for free products imply that P is standard for R G 1 . Thus, P can only be non-standard if w is the empty word, i.e. if P contains a basis element of N , which of course implies

P = N . Now if Φ : Ind G 1 * G 2 G 1 (N ) → Ind G 1 * G 2 G 1 (N ) is an isomorphism of R G 1 * G 2 -
modules, it can also be seen as an isomorphism of R G 1 -modules. This isomorphism preserves standard modules so it must also send N isomorphically to N , hence the result.

For clarity, the proof that torsion actions of a free product are always induced from one of the factors will be split into two parts. First we will work at the level of fusion modules and then recast the proof in the setting of module C*-categories. The argument is inspired by that of [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]Thm 1.25]. Proposition 3.4. Let M be a torsion R G 1 * G 2 -module. Then, it is induced from a torsion module of one of the factors.

Proof. Let e be a basis element of M and let N e 1 (resp. N e 2 ) denote the sub-R G 1 -module (resp. the sub-R G 2module) of M generated by e. The proof of [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]Thm 1.25] shows that if both N e 1 and N e 2 are standard for all e, then M is standard. In particular, it is induced from the standard module of either

R G 1 or R G 2 .
Let us assume that M is not standard and let e be such that N e 1 is not standard (the case where N e 2 is not standard is similar). There is a natural candidate for the isomorphism : for a word

w ∈ Irr(G 1 * G 2 )
ending in Irr(G 2 ) and a basis element j of N e 1 , it should send the basis element wj of

Ind G 1 * G 2 G 1
(N e 1 ) to w ⊗ j. However, this only makes sense if w ⊗ j is also a basis element. Let us prove this by induction on the length of w, with the following induction hypothesis :

H k : "Let w = γ k • • • γ 1 ∈ Irr(G 1 * G 2 )
be a word ending in Irr(G 2 ) and let j be a basis element of N e 1 . Then, w ⊗ j is a basis element."

For k = 1, assume that there exists β 2 ∈ Irr(G 2 ) \ {ε 2 } such that j ⊂ β 2 ⊗ j and let β 1 be a nontrivial element in Stab(j) ∩ Irr(G 1 ), which exists by Lemma 3.2 because N e 1 is not standard. Then, for any integer l, (β 1 β 2 ) l ∈ Irr(G 1 * G 2 ) is a non-trivial stabilizer of j. This yields innitely many stabilizers, contradicting coniteness. Thus, Stab(j) ∩ Irr(G 2 ) = {ε 2 }. This implies by assumption that the R G 2 -submodule generated by j is standard and the isomorphism must send j to ε 2 . In particular, γ 1 ⊗ j is a basis element for all γ 1 ∈ Irr(G 2 ) and H 1 holds.

Assume now H k for some k 1 and let

γ k • • • γ 1 ∈ Irr(G 1 * G 2 )
be a word ending in Irr(G 2 ). Let us assume for simplicity that γ k ∈ Irr(G 2 ) \ {ε 2 }, the other case being similar. Set j = γ k • • • γ 1 ⊗ j which is a basis element by H k . By the same argument as before, Stab(j ) ∩ Irr(G 1 ) = {ε 1 } so that N j 1 is standard for R G 1 with an isomorphism sending j to the trivial representation. In particular, for any γ k+1 ∈ Irr(G 1 ) \ {ε 1 }, γ k+1 ⊗ j is a basis element and H k+1 holds.

We can now nish the proof. Let

Φ : Ind G 1 * G 2 G 1 (N e 1 ) → M
be the map sending wj to w ⊗ j. This is a surjective module homomorphism by connectedness. Let w = w be words ending in Irr(G 2 ) and let j, j be basis elements in N e 1 such that Φ(wj) = Φ(w j ). Then,

j ⊂ (w ⊗ w) ⊗ j.
Observe that w ⊗ w is a sum of non-empty words starting and ending in Irr(G 2 ) \ {ε 2 }. In particular, there

exists w ∈ Irr(G 1 * G 2 ) ending in Irr(G 2 ) such that j ⊂ w ⊗ j.
Since we have proved that w ⊗ j is a basis element, we get j = w ⊗ j. Now, if β, β ∈ Irr(G 1 ) \ {ε 1 } stabilize respectively j and j , we get for any l ∈ N a stabilizer (w β w β) l of j, contradicting coniteness. Thus, w = w , j = j and Φ is faithful.

We can now state and prove the main result of this subsection.

Theorem 3.5. Let G 1 and G 2 be two compact quantum groups. Then, up to equivariant Morita equivalence there is a one-to-one correspondance between torsion actions of G 1 * G 2 and torsion actions induced from G 1 or G 2 .

Proof. As in [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]Thm 3.16], it suces to recast the previous proof in the setting of module C*-categories. Let (A, α) be a torsion action of G 1 * G 2 and consider the associated C*-module category (C, ⊗). For any irreducible object X in C, we can consider the module C*-categories C X 1 and C X 2 generated by X and the action of the representation categories of G 1 and G 2 respectively. By Proposition 3.4, there is an X such that one of them (say C X 1 ) has a non-standard fusion module if (A, α) is not trivial. By [1, Lem 3.10], the module C*-category is equivalent to the one of the trivial action if its associated fusion module is standard. The same reasoning as for fusion modules therefore yields an isomorphism between C and the module C*-category induced from C X 1 (by which we mean the module C*-category obtained by induction of all the objects of C X 1 ). To conclude, note that by the general results of [START_REF] Commer | Tannaka-Krein duality for compact quantum homogeneous spaces I. General theory[END_REF], there is a torsion action (A , α ) of G 1 such that the associated module C*-category is C X 1 . Thus, C is equivalent to the module C*-category associated to

Ind G 1 * G 2 G 1
(α ) and again by [START_REF] Commer | Tannaka-Krein duality for compact quantum homogeneous spaces I. General theory[END_REF] the actions are equivariantly Morita equivalent. Consider now two induced torsion actions which are equivariantly Morita equivalent. If they are induced from dierent factors, then the fusion module of the one induced from G 2 is a direct sum of standard modules when restricted to G 1 . Thus, the fusion module associated to the one induced from G 1 is isomorphic to a direct sum of standard module for R G 1 . Since all its submodules are also standard for R G 2 , we conclude by [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]Thm 1.25] that the actions are trivial. This leaves us with the case of two torsion actions (A, α) and

(A , α ) of say G 1 such that Ind G 1 * G 2 G 1 (α) and Ind G 1 * G 2 G 1
(α ) are equivariantly Morita equivalent. The same reasoning as in Lemma 3.3 shows that in both associated module C*-categories C and C , the module C*-subcategory coming from the original action is the only one to be non-trivial over the representation category of G 1 . The equivalence of categories must therefore restrict to an equivalence between these sub-categories and we conclude by [START_REF] Commer | Tannaka-Krein duality for compact quantum homogeneous spaces I. General theory[END_REF].

3.2. Free wreath product. In [START_REF] Bichon | Free wreath product by the quantum permutation group[END_REF], J. Bichon introduced a "free" version of the wreath product construction which produces, out of a compact quantum group G and a quantum permutation group S + N , their free wreath product G * S + N . This gives many examples of compact quantum groups and in particular the so-called quantum reection groups when G is the dual of a cyclic group. Once again, we will not give the denition of these objects since it is not necessary for our purpose. All we need is to understand their fusion ring, which was described by F. Lemeux and P. Tarrago in [START_REF] Lemeux | Free wreath product quantum groups : the monoidal category, approximation properties and free probability[END_REF]. Consider the free monoid F over Irr(G). Then, Z F is an abelian group. Moreover, we can dene an involution on it by setting

γ 1 • • • γ k = γ k • • • γ 1 .
We can also use the fusion rules of G to dene a ring structure on Z F by

(γ 1 • • • γ k ) ⊗ (γ 1 • • • γ n ) = γ 1 • • • γ k γ 1 • • • γ n ⊕   β⊂γ k ⊗γ 1 γ 1 • • • γ k-1 βγ 2 • • • γ n   ⊕ δ γ k ,γ 1 (γ 1 • • • γ k-1 ) ⊗ (γ 2 • • • γ n ).
If N 4, the based ring obtained in this way is the fusion ring of G * S + N . The idea to study torsion in G * S +

N is to embed it into a compact quantum group whose torsion is better understood, namely a free product. Such an embedding need not exist in general, but it always does once we consider a monoidally equivalent compact quantum group, thanks to the following result of F. Lemeux and P. Tarrago in [START_REF] Lemeux | Free wreath product quantum groups : the monoidal category, approximation properties and free probability[END_REF]. From now on, let us denote by u 1 the fundamental representation of SU q (2).

Theorem 3.6 (Lemeux-Tarrago). Let G be a compact quantum group and let H q be the compact quantum subgroup of G * SU q (2) generated by {u 1 αu 1 | α ∈ Irr(G)}. If N 4, then there exists 0 < |q| < 1 such that H q is monoidally equivalent to G * S + N . Remark 3.7. P. Fima and L. Pittau introduced in [START_REF] Fima | The free wreath product of a compact quantum group by a quantum automorphism group[END_REF] the free wreath product of a compact quantum group by any quantum automorphism group of a nite-dimensional C*-algebra B preserving a given state and proved a similar monoidal equivalence. In fact, H q is nothing but G * SO q (3) in the sense of [START_REF] Fima | The free wreath product of a compact quantum group by a quantum automorphism group[END_REF]Def 2.6].

We will therefore classify the torsion actions of H q . The point is that any torsion action of H q induces a torsion action of G * SU q (2). Since SU q (2) is known to be torsion-free by [20, Prop 3.2] (see also [START_REF] Arano | Torsion-freeness for fusion rings and tensor C*-categories[END_REF]Prop 1.23] for a combinatorial proof), the torsion actions of the free product are induced from torsion actions of G by Theorem 3.5. For convenience, let us give an explicit description of the inclusion Λ : R Hq ⊂ R G * SUq [START_REF] Baaj | Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres[END_REF] . Let us denote by u n the n-th irreducible representation of SU q (2) and let w be a word in the free monoid over Irr(G).

•

If w = ε n G , then Λ(w) = u 2n , • if w = β 1 • • • β n with β i = ε G for all 1 i n, then Λ(w) = u 1 β 1 u 2 β 2 u 2 • • • u 2 β n u 1 , • otherwise, we can write w as w = ε n 1 G w 1 ε n 2 G w 2 • • • w k ε n k+1 G
where each w i does not contain ε G . Let us denote by Λ(w i ) the same expression as Λ(w i ) except that the rst and last u 1 's are removed. Then,

Λ(w) = u 2n 1 +1 Λ(w 1 )u 2n 2 +2 Λ(w 2 ) • • • u 2n k +2 Λ(w k )u 2n k+1 +1 .
One can easily check that these expressions are compatible with the tensor products and that Λ is an isomorphism. What we now have to understand is how torsion modules of G * SU q (2) behave when we only consider the action of H q . Lemma 3.8. Let N be a torsion module for R G and let M = Ind G * SUq(2) G (N ). Then, M contains a unique non-standard torsion module for R Hq .

Proof. Let j be a basis element of N , let w ∈ Irr(G * SU q (2)) be a word ending in Irr(SU q (2)) and let N (wj) be the sub-R Hq -module generated by the basis element wj of M . We will denote by w a word of minimal length such that w j ∈ N (wj). If w = ∅, then for any word w ending in Irr(SU q (2)) \ {ε}, Λ(w ) ⊗ j = Λ(w )j so that N (j) is standard. If w starts in Irr(G), then again Λ(w ) ⊗ w j = Λ(w )w j for any word w ending in Irr(SU q (2)) \ {ε}, so that N (wj) = N (w j) is standard. Let us therefore assume

that w = u n 0 β 1 • • • β k u n k . If n 0 > 1, then since u 2 ∈ Irr(H q ) we see that u n 0 -2 β 1 • • • β k u n k j ∈ N (wj)
and we can assume that n 0 = 1. If k > 0, then tensoring by u 1 β 1 u 1 we see that

u n 1 β 2 • • • β k u n k j ∈ N (wj),
contradicting the minimality of w . Thus, w = u 1 . Note that for any basis element j of N , there exists β ∈ Irr(G) such that j ⊂ β ⊗ j, therefore u 1 j ⊂ u 1 βu 1 ⊗ u 1 j. This implies that N (u 1 j) = N (u 1 j ) and we therefore get only one torsion module, denoted by N u 1 .

To conclude, we now have to show that N u 1 is not standard. But this is clear since u 2 is a non-trivial stabilizer of all its basis elements.

As for free products, the statement for actions can be deduced from this algebraic result. Theorem 3.9. Let G be a compact quantum group and let N 4. Then, the equivariant Morita equivalence classes of non-trivial torsion actions of G * S + N are in one-to-one correspondence with all the equivariant Morita equivalence classes of torsion actions of G.

Proof. Let (A, α) be a torsion action of H q . By Theorem 3.5, Ind G * SUq (2) Hq (α) is equivariantly Morita equivalent to an action induced from G. By Lemma 3.8, the restriction of such an action to H q has exactly one nontrivial summand. Thus, this summand is equivariantly Morita equivalent to (A, α). Moreover, if two torsion actions of H q are equivariantly Morita equivalent, then the same holds for their induction to G * SU q (2), so that by Theorem 3.5 the original actions of G are also equivariantly Morita equivalent.

In particular, a free wreath product is never torsion-free since the trivial action of G gives rise to a nontrivial torsion action of G * S + N . Let us describe explicitly this action. Consider the quantum subgroup of H q generated by u 2 , which is isomorphic to SO q (3). It is known that it admits a projective action α q on M 2 (C) which is a non-trivial torsion action. It was proven in [START_REF]On the structure of quantum automorphism groups[END_REF]Lem 4.4] that this is the only non-trivial torsion action of SO q (3) up to equivariant Morita equivalence. Note that this action becomes trivial when induced to SU q (2). Similarly, by Theorem 3.9 Ind Hq SOq(3) (α q ) is a nontrivial torsion action whose induction to G * SU q (2) is trivial. Under the monoidal equivalence, (M 2 (C), α q ) becomes the dening action of C(S + N ) on C N . Inducing this action to G * S + N yields a nontrivial torsion action (C N , α N ) which is precisely the one obtained from the trivial action of G. If G is torsion-free, this is the only source of torsion in the free wreath product : Corollary 3.10. Let G be a torsion-free compact quantum group and let N 4. Then, (C N , α N ) is the only non-trivial torsion action of G * S + N up to equivariant Morita equivalence.

3.3. Free complexication. Before ending this section, we want to investigate one last construction called free complexication and introduced by T. Banica in [START_REF] Banica | A note on free quantum groups[END_REF]. This is an important source of free unitary quantum groups and may therefore be an interesting object from the point of view of K-theory computations. However, it turns out that in that case, understanding torsion is more dicult than for free wreath products. We will give some general results and then work out the particular case of the complexied hyperoctahedral quantum group H + N . We start by recalling the denition of free complexication, which is slightly dierent from the previous ones because it does not only take a compact quantum group as an argument but also a distinguished representation. A representation u of a compact quantum group G is said to be a fundamental representation of G if any irreducible representation is contained up to equivalence in a tensor power of u and its conjugate. Denition 3.11. A compact matrix quantum group is a pair (G, u) where G is a compact quantum group and u is a fundamental representation of G. If moreover u = u, then (G, u) is said to be orthogonal.

Let S 1 denote the compact group of complex numbers of modulus 1 and consider the free product G * S 1 . If z denotes the identity representation of S 1 , then u = uz is a representation of G * S 1 and we can consider the quantum subgroup G generated by this representation. This yields a new compact matrix quantum group ( G, u) called the free complexication of (G, u).

Before studying the behaviour of torsion actions under this construction, let us prove a result concerning divisibility. The notion of divisible quantum subgroup was introduced in [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF] to study the Baum-Connes conjecture for the free unitary quantum groups U + F . There are several equivalent denitions but we only give the one which will be useful for us. Let G and H be compact quantum groups such that C(H) ⊂ C(G). Then, Irr(H) embeds into Irr(G) and we can dene an equivalence relation on Irr(G) by setting β ∼ β if there exists γ ∈ Irr(H) such that β ⊂ β ⊗ γ. Denition 3.12. The quantum group H is said to be divisible in G if for any class X ∈ Irr(G)/ Irr(H), there exists a representative β of X such that for all γ ∈ Irr(H), β ⊗ γ is irreducible.

It was proven in [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF]Prop 4.3

] that U + F is divisible in O + F * S 1
, where O + F denotes the corresponding free orthogonal quantum group. We will extend this result to arbitrary compact quantum groups provided that the fundamental representation is orthogonal. To do so, we rst introduce an important subgroup of (G, u). Denition 3.13. Let (G, u) be an orthogonal compact matrix quantum group. The compact quantum subgroup G ev of G generated by u ⊗ u is called the even part of (G, u).

Equivalently, the irreducible representations of G ev are exactly the irreducible representations of G which are subrepresentations of u ⊗2k for some k ∈ N. Note that because u ⊗ u ⊂ (uz) ⊗ (zu), C(G ev ) ⊂ C( G). Proposition 3.14. Let (G, u) be an orthogonal compact matrix quantum group. Then, the free complexi-

cation G is divisible in G * S 1 . Proof. If C(G ev ) = C(G), then C(G) ⊂ C( G). This implies that z ⊂ u ⊗ (uz) ∈ Irr(C( G))
and eventually that G = G * S 1 . In that case, the result is trivial.

Let us therefore assume that G ev = G and describe the irreducible representations of G inside G * S 1 . Following the proof of [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF]Prop 4.3], we set, for ∈ {-1, 1}, [ ] -= min( , 0) and [ ] + = max( , 0). Let W be the subset of Irr(G * S 1 ) consisting in words of the form

z [ 0 ] -β 1 z 1 • • • z p-1 β p z [ p]+ with i+1 = -i if β i ∈ Irr(G ev ) and i+1 = i if β i ∈ Irr(G) \ Irr(G ev ).
By denition, W is stable under taking tensor products and contragredients. Moreover, it contains uz so that Irr( G) ⊂ W . One can prove that the reverse inclusion also holds but we will not need it hereafter.

Let now X ∈ Irr(G * S 1 )/ Irr( G) and let w be a representative of X of minimal length. If w is the trivial representation then we are done. Assume therefore that w has length 1. If w = z k with k ∈ Z * , then the result is also clear. If w = β for some representation β ∈ Irr(G), we have two cases : if β ∈ Irr(G ev ) then tensoring with β yields ε ∈ X. Otherwise, tensoring by βz yields z ∈ X and we can again conclude. Let us eventually assume that w has length at least 2. If it ends with z k for some k = 1, the result is clear. Otherwise, we can as before reduce the length of w, hence the result.

A consequence of Proposition 3.14 is that if G is torsion free and has the strong Baum-Connes property, then so does G. This follows from [1, Prop 1.28] and [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF]Thm 6.6] but does not settle the general case. We therefore have to investigate how torsion may pass to the free complexication. Surprisingly, the answer is not as clear as before because it involves the relationship between modules on R G and modules on R Gev .

Let us x an orthogonal compact matrix quantum group (G, u) once and for all. For convenience, we will say that an irreducible representation β ∈ Irr(G) is even if it is in Irr(G ev ) and odd otherwise. Of course, everything is trivial if G ev = G. We will therefore assume from now on that this is not the case, i.e. that there exist odd representations. Given a torsion module N on R G , we want to understand how it splits over G ev and in particular how many connected components there will be. This can be captured by the following equivalence relation on the basis elements J of N : j ∼ j if there exists β ∈ Irr(G ev ) such that j ⊂ β ⊗ j . This is an equivalence relation since it means that the two basis elements generate the same R Gev -submodule. Intuitively, G ev is an "index two" subgroup of G and we may therefore expect that the number of equivalence classes is at most two. Proposition 3.15. There are at most two equivalence classes. Moreover, the following are equivalent :

(1) All basis elements have an odd stabilizer.

(2) There exists a basis element with an odd stabilizer.

(3) There is only one equivalence class, i.e. the module N is R Gev -connected. Proof. Let j 1 , j 2 , j 3 be basis elements such that the rst two are not equivalent to j 3 . By connectedness, there exists β 1 , β 2 ∈ Irr(G) such that j 1 ⊂ β 1 ⊗ j 3 and j 3 ⊂ β 2 ⊗ j 2 . Moreover, β 1 and β 2 are odd by assumption, so that any subrepresentation of

β 1 ⊗ β 2 is even. Since j 1 ⊂ (β 1 ⊗ β 2 ) ⊗ j 3 , we get j 1 ∼ j 3 .
As for the equivalence of the statements, we will prove that each one implies the next one. The rst implication is trivial. For the second one, let j be a basis element with an odd stabilizer β, let j be another basis element and let γ ∈ Irr(G) satisfy j ⊂ γ ⊗ j . If γ is even, then we are done. Otherwise, j ⊂ (β ⊗ γ) ⊗ j and the tensor product contains only even representations, hence the result. For the third implication, let j be a basis element and let j ⊂ u ⊗ j. Then, there exists an even representation γ such that j ⊂ γ ⊗ j , so that γ ⊗ u contains an odd stabilizer of j.

There is always at least one module not satisfying the above equivalent conditions, namely the standard module of R G , since ε then has trivial stabilizer. More precisely, as R Gev -modules we have R G = R Gev ⊕R odd , where R odd is the span of all odd representations (which is also the submodule generated by u). However, there can also be other non-standard R G -modules whithout odd stabilizers. For instance, the action of R Z 4 on the two-dimensional module Z.j 0 ⊕ Z.j 1 where even elements stabilize each point while odd elements exchange them.

We can now link this with torsion modules on the free complexication. To do this, rst note that any torsion module of R G can be induced to a torsion module of G * S 1 and this induced module is by Theorem 3.5 isomorphic to one induced from R G . Thus, we can focus on modules coming from G. Proposition 3.16. Let N be a non-standard torsion module of R G and let M = Ind G * S 1 G (N ). Then, M contains one or two non-standard torsion R G -submodules, denoted by P and P . Moreover,

P = P if and only if N is R Gev -connected.
Proof. Recall that the basis of M consists in elements of the form wj with w a word on {z k , k = 0} ∪ Irr(G) \ {ε G } not ending in Irr(G). Let P be a R G -submodule and consider a word w and a basis element j of N such that wj ∈ P and w has minimal length. We have several cases :

• If w starts with an element γ of Irr(G ev ) ⊂ Irr( G), then letting γ act we see that we can reduce the length of w, contradicting minimality. • If w starts with z k with k = -1, then the description of the inclusion Irr( G) ⊂ Irr(G * S 1 ) above implies that corresponding module is standard. • If w starts with an element β of Irr(G) \ Irr(G ev ) and is of the form βz k w , then tensoring with z -1 β

shows that we can replace w by z k-1 w , contradicting minimality.

• If w = β ∈ Irr(G) \ Irr(G ev ),
tensoring by z -1 β shows that we can assume w = z -1 . Let then j = j be such that there is γ ∈ Irr(G) \ Irr(G ev ) such that j ⊂ γ ⊗ j. Then j ⊂ γz ⊗ z -1 j, contradicting minimality. As a conclusion, the word w must be empty and we have to consider submodules generated by basis elements of N . Let j be such a basis element and let P j be the R G -submodule generated by j. Because R Gev ⊂ R G , if j ∼ j then P j = P j . Reciprocally, any word w ∈ Irr( G) of length at most two contains a power of z, hence cannot connect two basis elements of N . Thus, by Proposition 3.15, all the sub-modules P j coincide if and only if N is connected as a R Gev -module. Otherwise, we get two distinct submodules P and P .

Assume that P and P are both standard. Then, M was the standard module of R G * S 1 , hence by Theorem 3.5 N was standard, contradicting the assumption. As a consequence, there is at least one non-standard torsion R G -submodule. The main problem with the previous statement is that when all the stabilizers are even, we get two nonstandard modules so that there may be more torsion in G than in G. It is moreover not clear whether the torsion modules obtained are isomorphic to one another, or isomorphic to those arising from other torsion modules. At least, in the particular case of H + N we can settle this issue. Recall that

H + N = Z 2 * S +
N is the free hyperoctahedral quantum group and H + N is its free complexication.

Corollary 3.17. The quantum group H + N has exactly two non-trivial torsion actions up to equivariant Morita equivalence.

Proof. By Theorem 3.9, H + N has two non-trivial torsion actions up to equivariant Morita equivalence. One of them comes from the trivial R Z 2 -module Z.j 0 . At the level of the free wreath product, it is generated by u 1 j 0 and this element has the odd stabilizer u 1 su 1 , where s denotes the generator of Z 2 seen as an irreducible representation. Thus, P = P in that case.

As for the second one, it comes from the standard module Z.j + ⊕ Zj -of Z 2 where s exchanges the two basis elements. In that case, all stabilizers are even so that P = P . However, there is a R H + N -equivariant automorphism θ of N given by u 1 j + → u 1 j -which restricts to an isomorphism between the two summands of the restriction of N to the even part of H + N . Thus, P is isomorphic to P . As before, the proof can be recast in the context of module categories to yield the result. Moreover, if two torsion actions are equivariantly Morita equivalent, so is their induction to the free product so that by Theorem 3.5 we can conclude that they come from the same torsion action of H + N .

3.4. Application to the Baum-Connes conjecture. Classifying the torsion actions of a discrete quantum group is a rst step towards the statement of a suitable form of the Baum-Connes conjecture. In this section we will give such a statement and prove it, essentially following [START_REF]On the structure of quantum automorphism groups[END_REF]. The argument involves duality for quantum groups and therefore requires some conventions. Given a compact quantum group G, we will denote by G its dual discrete quantum group. The C*-algebra C * ( G) is then isomorphic to C(G) and the Baum-Connes property concerns G. More precisely, it concerns the category KK G of G-C*-algebras with morphisms given by the equivariant KK-groups. By Baaj-Skandalis duality [START_REF] Baaj | Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres[END_REF], A → G A implements an equivalence of categories KK G → KK G so that we can chose either the compact or discrete point of view. Let now G be a compact quantum group and let C(H q ) ⊂ C(G * SU q (2)) be the compact quantum group dened in Theorem 3.6. Monoidal equivalence implements an equivalence of categories KK G * S + N → KK Hq so that it is enough to study H q or its dual. We now introduce some terminology from [START_REF] Meyer | Homological algebra in bivariant K-theory and other triangulated categories I[END_REF]. Let T Hq ⊂ KK Hq be the full subcategory generated by all H q -C*-algebras of the form B ⊗ C where B is a torsion action of H q and the action is only on the rst tensor factor. It is known by [START_REF] Meyer | The Baum-Connes conjecture via localisation of categories[END_REF] that KK Hq is a triangulated category so that we may consider the localizing subcategory T Hq ⊂ KK Hq generated by T Hq and denote by CI Hq ⊂ KK Hq the corresponding subcategory. We will say that the discrete quantum group H q satises the strong Baum-Connes property if CI Hq is equal to the whole category KK Hq .

We will prove this property in the case of a torsion-free G. The reason for this assumption is that there is only one non-trivial torsion action, which can moreover be nicely described. Let S be the subset of irreducible representations of G * SU q (2) generated by the action of Irr(H q ) on u 1 and let us consider inside C * (G * SU q (2)) the closure A q of the direct sum of coecients of representations in S. Then, C * (G * SU q (2)) splits as a H q -C*-algebra into a direct sum of A q and copies of C * (H q ) by Theorem 3.9. We rst want to indentify A q as a crossed product, i.e. an image under Baaj-Skandalis duality. Lemma 3.18. Let (M 2 (C), α q ) be the torsion action of H q dened in Subsection 3.2. Then, A q is equivariantly Morita equivalent to H q M 2 (C).

Proof. By Theorem 3.9, the image of A q under Baaj-Skandalis duality is the only non-trivial summand in the restriction of the trivial action of G * SU q (2) to H q , hence it is equivariantly Morita equivalent to (M 2 (C), α q ). Taking duals again yields the result. Note that one can also produce an explicit imprimitivity bimodule by mimicking the proof of [START_REF]On the structure of quantum automorphism groups[END_REF]Lem 5.1].

We are now ready to prove our main result concerning the strong Baum-Connes property. Proof. By monoidal equivalence, it is enough to prove the result for H q and even for H q . So let B be any

H q -C*-algebra and let B = Ind G * SU q (2)
Hq (B) be its induction. Because

T G * SUq(2) = KK G * SUq(2)
by [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF] and G * SU q (2) is torsion-free by Theorem 3.5, B is in the localizing subcategory generated by elements of the form C * (G * SU q (2)) ⊗ D where the action is only on the rst factor. Using the fact that

C * (G * SU q (2)) = A ⊕ C * (H q ) ⊕N , we see that B = Res G * SU q (2)
Hq (B ) is in CI Hq . Moreover, B is a direct summand in B as explained in the proof of [START_REF]On the structure of quantum automorphism groups[END_REF]Thm 5.2] In particular, all elements of CI Hq are proper almost homogeneous tensored by a trivial action by Lemma 3.18. This property passes to the generated subcategory, that is to say to all of KK Hq by Theorem 3.19. Because the action of a discrete quantum group on a proper almost homogeneous C*-algebra is amenable by [START_REF]On the structure of quantum automorphism groups[END_REF]Lem 4.6], we conclude that H q is K-amenable. The property of being proper almost homogeneous is stable under monoidal equivalence, thus the same reasoning works for G * S + N , yielding Corollary 3.22. Let G be a torsion-free compact quantum group satisfying the strong Baum-Connes property.

Then, the dual of G * S + N is K-amenable. Note that because all our arguments carry through monoidal equivalence, the result holds for any free wreath product by an arbitrary quantum automorphism group of a nite-dimensional C*-algebra with a distinguished state in the sense of [START_REF] Fima | The free wreath product of a compact quantum group by a quantum automorphism group[END_REF].

K-theory computations

From now on, G denotes a torsion-free compact quantum group with the strong Baum-Connes property and we set G = G * SU q (2). We will use the previous results to compute the K-theory of the C*-algebras associated to the compact quantum group H q = G * SO q (3). The point here is that the corresponding C*-algebra sits inside C(G) so that we can use restriction to produce an exact sequence for the K-theory of H q from one for G. More precisely, assume that we have a projective resolution of C of length one in

KK G 0 -→ B 1 -→ B 0 -→ C -→ 0.
The general theory of [START_REF] Meyer | The Baum-Connes conjecture via localisation of categories[END_REF] then implies that the Dirac element C sits in an exact triangle (1)

B 1 -→ B 0 -→ C -→ ΣB 1 .
The image of this triangle in KK Hq is still exact, hence yields a six-term exact sequence in K-theory, from which we can compute the K-theory groups of C * ( H q ) = C(H q ). This strategy was used in [START_REF]Quantum SU (2) and the Baum-Connes conjecture[END_REF]Sec 5] for quantum automorphism groups of matrices, which is the case where G is trivial. In the general case, the computations can be split into two parts, one of which only depends on SU q (2) and is very similar to those of [START_REF]Quantum SU (2) and the Baum-Connes conjecture[END_REF]Sec 5]. We will therefore deal with this rst part separately.

4.1. Preliminary computations. We denote by u the fundamental representation of SU q (2) and by p u the minimal central projection associated to u in C 0 ( G). Dene, following [START_REF] Voigt | The Baum-Connes conjecture for free orthogonal quantum groups[END_REF], a KK-theory element T u ∈ KK Hq (C 0 ( G), C 0 ( G)) as the composition of

(id ⊗p u ) • ∆ : C 0 ( G) → C 0 ( G) ⊗ C 0 ( G) u with the canonical equivariant Morita equivalence C 0 ( G) ⊗ C 0 ( G) u ∼ C 0 ( G). More explicitly, this is the Kasparov module C 0 ( G) C 0 ( G) ⊗ H u C 0 ( G)
where the right action is by multiplication on the rst tensor and the left action is through

(id ⊗p u ) • ∆ : C 0 ( G) → C 0 ( G) ⊗ B(H u ).
We want to understand the image of T u under the descent morphism

j : KK Hq (C 0 ( G), C 0 ( G)) → KK( H q C 0 ( G), H q C 0 ( G)).
Let us rst decompose C 0 ( G) as an H q -algebra. For a word w on Irr(G) \ {ε G } Irr(SU q (2)) \ {ε SUq(2) } which is either empty or starts in Irr(G), we denote by J w the set of irreducible representations which are contained in a tensor product of an irreducible representation of H q by w and we write

A w = x∈Jw B(H x ),
where is a c 0 -sum. According to the results of subsection 3.2, A w is equivariantly Morita equivalent to C 0 ( H q ). In fact, it is even isomorphic to C 0 ( H q ) ⊗ B(H w ) with the trivial action on the second factor. Moreover,

C 0 ( G) = A u ⊕ w A w ,
where A u is the sum of the blocks obtained by tensoring representations of H q by u. The KK-element T u therefore splits as a direct sum and we will study each summand separately. Note that the crossedproduct of each A w by H q is isomorphic to the compact operators. The same holds for A u since the crossed product, being a non-trivial torsion action, must be equivariantly Morita equivalent to M 2 (C), hence Morita equivalent to C. Thus, after splitting and applying the descent morphism to each component, the result can be described as a matrix acting on a free Z-module of innite rank. Let us denote by e w the unit of the copy of Z corresponding to A w and by e u the one corresponding to A u . Proposition 4.1. The element ∂ u = j(T u ) acts as follows on the basis :

• ∂ u (e wu k ) = e wu k+1 + e wu k-1 for w = ∅ ending in Irr(G), • ∂ u (e w ) = e wu for w = ∅ ending in Irr(G), • ∂ u (e ∅ ) = 2e u and ∂ u (e u ) = 2e ∅ .
Proof. If a ∈ B(H x ), then (id ⊗p u ) • ∆(a) belongs the sum of the tensor products B(H y ) ⊗ B(H u ) with y such that x ⊂ y ⊗ u. We therefore have three cases :

• If x ∈ J wu k , then y ∈ J wu k+1 or y ∈ J wu k-1 .

• If x ∈ J w with w = ∅ ending in Irr(G), then y ∈ J wu .

• If x ∈ J u then y ∈ J ∅ and if x ∈ J ∅ then y ∈ J u . Let us concentrate on the rst case. We have to understand the module

Hq C 0 ( Hq)⊗B(H wu k ) H q C 0 ( H q ) ⊗ B(H wu k+1 ) ⊗ H u Hq C 0 ( Hq)⊗B(H wu k+1 )
where the right action is multiplication on the rst two tensors and the left action is multiplication for the crossed-product and (p wu k+1 ⊗p u )• ∆ for B(H wu k ). The crossed-product appearing here is by Takesaki-Takai duality isomorphic to the compact operators so that we can remove it. Applying then Morita equivalence on the right action, we get

B(H wu k ) (B(H wu k+1 ) B(H wu k+1 ) H wu k+1 ⊗ H u ) C = B(H wu k ) (H wu k+1 ⊗ H u ) C ,
with the left action given simply by the embedding of B(H wu k ) as a corner in B(H wu k+1 ⊗ H u ). Applying now Morita equivalence on the left action yields

C (H wu k B(H wu k ) (H wu k+1 ⊗ H u )) C .
Recall that H wu k+1 ⊗ H u = H wu k ⊕ H wu k+2 and that the tensor product with the second component vanishes since B(H wu k ) acts on it by 0. We are therefore left with

H wu k B(H wu k ) H wu k which is 1 ∈ Z = KK(C, C).
The same computation works for the term wu k-1 .

If w = ∅ ends in Irr(G), the previous computation still works and is even simpler since there is only one possible component for the right-hand side. As for the third case, it is clear that e ∅ and e u are exchanged. Moreover, the action of j(T u ) on Ze ∅ ⊕ Ze u does not depend on the compact quantum group G so that it is enough to do the computation when G is trivial. In that case, H q = SO q (3) and the result was proven in [21, Sec 5].

We will need later on to know the image of the linear map d u = ∂ u -2 id, which is easily computed. For convenience, let us set E w = span{e wu k | k ∈ N} for w = ∅ and E u = Ze ∅ ⊕ Ze u . Lemma 4.2. The image of E u under d u is 2Z(e ∅ -e u ). Moreover, let (a k ) k∈N be the sequence dened by a 0 = 2, a 1 = 3 and a k+1 = 2a k -a k-1 . Then, the image of E w is the free module spanned by the vectors k = e wu k+1 -a k e w . Proof. The rst point is clear. For the second point, we will proceed by induction, showing that the image of the span of (d u (e wu l )) l for 0 l k equals the span of ( l ) l . For k = 0, we indeed have d u (e w ) = 0 and for k = 1 a straightforward computation yields

d u (e wu ) = e wu 2 -3e w -2 0 = 1 -2 0 .
If we now assume H k for k 1, we get

d u (e wu k+1 ) = e wu k+2 + e wu k -2e wu k+1 = e wu k+2 + ( k-1 + a k-1 e w ) -2( k + a k e w ) = e wu k+2 -(2a k -a k-1 )e w + k-1 -2 k = k+1 + k-1 -2 k
Since the family ( k ) k is free, the result follows.

4.2. Free orthogonal quantum groups. The rst computation will be for G = O + n . Let us denote by v its fundamental representation and let T v ∈ KK Hq (C 0 ( G), C 0 ( G)) be dened in the same way as T u . We will denote its image under the descent morphism by ∂ v and set d v = ∂ v -n id. Consider the homological ideal J = ker(Res : KK G → KK). It was proven in [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF] that the complex (2)

0 -→ C 0 ( G) ⊕2 δ -→ C 0 ( G) λ -→ C -→ 0, where δ = (T u -dim(u) id) ⊕ (T v -dim(v) id)
and λ is the regular representation, is a J-projective resolution in KK G . It therefore yields an exact triangle in KK Hq containing C. Applying the functor K( H q •) leads to the six-term exact sequence

( w Z w ) ⊕ Z u / / K 0 (C(H q )) / / 0 (( w Z w ) ⊕ Z u ) ⊕2 du⊕dv O O K 1 (C(H q )) o o 0 o o
and the K-theory groups are given by the kernel and coimage of d u ⊕ d v . To compute these spaces, we need an analogue of Proposition 4.1 for ∂ v . The map d u ⊕ d v acts on two copies of (⊕ w Z w ) ⊕ Z u and for convenience we will denote by (f w ) w the basis of the second copy.

Proposition 4.3. The element ∂ v = j(T v ) acts as follows on the basis :

• ∂ v (f wv k ) = f wv k+1 + f wv k-1 ,
• ∂ v (f w ) = f wv for w ending in Irr(SU q (2)) or w = ∅.

• ∂ v (f u ) = nf u . Proof. The rst two cases follow from the same computations as in Proposition 4.1. As for the third one, rst note that the only representation y such that u ⊂ y ⊗ v is uv and that uv ∈ J u since uv ⊂ uvu ⊗ u. Thus, ∂ v (f u ) ∈ Zf u . Before applying the descent map j, we are considering the Kasparov module

Au (A u ⊗ H v ) Au
with the left action given by (id ⊗p v ) • ∆ and the right one given by multiplication on the rst tensor. Taking crossed-products we get Hq Au ( H q A u ⊗ H v ) Hq Au By Lemma 3.18, this yields the same KK-theory element as

M 2 (C) (M 2 (C) ⊗ H v ) M 2 (C)
where the right action is multiplication on the rst tensor and all we have to know on the left action is that it is faithful. Applying Morita equivalence on the right, we get

M 2 (C) (C 2 ⊗ H v ) C = C ([C 2 ⊗ H v ] ⊗ M 2 (C) C 2 ) C .
By faithfulness of the representation, the latter Hilbert space has dimension n, hence the result.

The problem now boils down to linear algebra in Z-modules. For clarity we rst determine the kernel. Proof. Consider an element x = λ u e u + µ u f u + w λ w e w + µ w f w and let L be the maximal length of the words appearing with non-zero coecient in this sum. If we assume that L > 1, then a word w of length L is either of the form w u k or w v k . In the rst case, let k 0 be maximal so that w = w u k 0 occurs in x. Then, Then,

(d u ⊕ d v )(x) = λ u (2e ∅ -2e u ) + 0 + λ ∅ (2e u -2e ∅ ) + µ ∅ (f v -nf ∅ ) = (2λ u -2λ ∅ )e ∅ + (2λ ∅ -2λ u )e u + µ ∅ (f v -nf ∅ ),
forcing λ u = λ ∅ and µ ∅ = 0.

We can now compute the K-theory of H q for G = O + n .

Proposition 4.5. Let G = O + n with n 2. Then, K 0 (C(H q )) = Z ⊕ Z 2 and K 1 (C(H q )) = Z 2 . Proof. The computation of K 1 (C(H q )) follows from Lemma 4.4. To compute K 0 (C(H q )), we rst have to compute the image of the two maps. For d u this was done in Lemma 4.2. For d v a similar computation shows that the image is spanned by the vectors η k = e wv k+1 -b k e w where w is a (possibly empty) word ending in Irr(SU q (2)) \ {ε by Lemma 4.2. Similarly, if w = w v k then p(e w ) = b k-1 p(e w ). Thus, the coimage is spanned by the vectors p(e w ) for w of length at most one. Since p(e v k ) = b k-1 p(e ∅ ), we even have to consider only p(e ∅ ) and p(e u ) so that the coimage has rank at most two. Since e u is not in the image of d v , the only relation between p(e ∅ ) and p(e u ) is the one coming from d u , namely 2p(e ∅ ) = 2p(e u ). Hence, the coimage is Z ⊕ Z 2 generated by p(e ∅ + e u ) and p(e ∅ -e u ).

4.3.

Free unitary quantum groups. The previous result can easily be extended to free unitary quantum groups and even to arbitrary free products of free orthogonal and free unitary quantum groups. Indeed, let

G = U + P 1 * • • • * U + P k * O + Q 1 * • • • * O + Q l
where P i ∈ GL m i (C) with m i 2 and Q j ∈ GL n j (C) with n j 2 satises Q j Q j = ± id. It is known from [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF] that if v i , u j denote the fundamental representations of U + P j and O + Q j respectively, then

δ = i (T v i -dim(v i ) id) ⊕ (T v i -dim(v i ) id) ⊕   j T u j -dim(u j ) id  
yields a projective resolution

0 -→ C 0 ( G) ⊕2k+l δ -→ C 0 ( G) λ -→ C -→ 0.
The same computations as before then yield

If Γ is a torsion-free group which is not free, then the above sequence does not work anymore. Indeed, any relation in Γ will produce elements in the kernel of d so that its injectivity fails. For instance if we consider Z 2 instead of F 2 , then d a -1 (a -ab) = d b -1 (aba -1 -ab).

One way to go round this problem could be to build a longer projective resolution. The price to pay then is that the exact sequence in K-theory will not be obtained directly but through a spectral sequence computation.

Theorem 3 .

 3 19. Let G be a torsion-free discrete quantum group satisfying the strong Baum-Connes property. Then, G * S + N satises the strong Baum-Connes property, i.e. T G * S + N = KK G * S + N .

Lemma 4 . 4 .

 44 The kernel of d u ⊕ d v is Z(e ∅ + e u ) ⊕ Zf u .

λ

  w e w u k 0 +1 -(d u ⊕ d v )(x)must be a linear combination of basis vectors not including e wu k 0 +1 , which is impossible if(d u ⊕ d v )(x) = 0. Similarly, if µ w = 0 then µ w f wv -(d u ⊕ d v )(x)yields a contradiction. The same argument works if w = w v k . If we assume now that L = 1 and w = v k we get the same contradiction. In conclusion, the sum only contains terms associated to w = u or w = ∅ and x = λ u e u + µ u f u + λ ∅ e ∅ + µ ∅ f ∅ .

  SUq(2) } and b k is dened by b 0 = n, b 1 = n 2 -1 and b k+1 = nb k -b k-1 . Let us denote by p the canonical surjection onto the coimage of d u ⊕ d v . If w = w u k is a word of length at least two, then p(e w ) = a k-1 p(e w )

  . Since triangulated subcategories are stable under retracts, B is in CI Hq and we conclude by Baaj-Skandalis duality. Remark 3.20. The idea of the proof, which comes from [22, Thm 5.2] yields a more general statement : if H ⊂ G and if the restriction functor Res G H preserves CI, then H satises the strong Baum-Connes property as soon as G does. We can also get another structure result for G * S + N , called K-amenability but this requires some terminology. Denition 3.21. A G-C*-algebra A is said to be proper almost homogeneous if it is equivariantly Morita equivalent to G B for some torsion action B of G.

Proposition 4.6. Let G be a free product of free orthogonal and unitary quantum groups as above. Then, K 0 (C(H q )) = Z ⊕ Z 2 and K 1 (C(H q )) = Z 2k+l+1 . 4.4. Classical free groups. Our third and last example will be free groups G = F n on nitely many generators. The case n = 1 is particularly interesting since the corresponding free wreath product Z * S + N is isomorphic to the quantum reection group H ∞+ N . The resolution given in [START_REF] Vergnioux | The K-theory of free quantum groups[END_REF] is not valid if we replace U + n by a free groups, but it is easy to modify it. Keeping the previous notations and denoting by a 1 , • • • , a n the generators of F n , we consider the complex

where δ = (T a 1 -id) ⊕ • • • ⊕ (T an -id) ⊕ (T u -2 id). We need to prove that this is a J-projective resolution, where J is the kernel of the restriction functor KK F N * SUq(2) → KK. By denition, it is enough to check exactness when applying the functor

n ⊕ d u and ε is the map induced by the dimension function.

Lemma 4.7. The diagram ( 3) is an exact sequence.

Proof. The surjectivity of ε is clear, let us prove that d is injective. We will denote by e k w the basis element corresponding to the representation w in the k-th copy of R G . Assume that x is a nite linear combination of such elements in the kernel of d and let w be a word of maximal length appearing in x. If it appears in the last component, then either w ends in F n and its image under d contains wu, or it ends with some u k and taking k maximal, its image under d contains the same word but ending with u k+1 . In both, cases, we get a contradiction by the same argument is in Lemma 4.4. We may therefore assume that w appears in one of the rst n components, say the rst one (the other cases being similar). If w ends with u k , then again we get a longer word after applying d which cannot be simplied. Thus, w = w γ for some γ ∈ F n and we may assume that the word length of γ is maximal. Then, still by the same argument γ must end with a 1 otherwise we would get a longer word. But then, d(wγ) = wγa -1

1 -wγ so that wγ appears in d(x). This means that there must be terms in x whose images simplies with wγ, but because we are working with free groups such a term must be of the form w(γa i ), contradicting the maximality of the word length of γ. As a conclusion, d is injective.

The vectors x w = w -dim(w) for a basis of the kernel of ε and we now have to compute their image in the quotient by the image of d (which is clearly contained in ker(ε)). Let us consider a non-empty word w. If w = w u k , the equality

(with the convention w u -1 = 0) yields

so that p(x w ) ∈ Zp(w u k-2 )⊕Zp(w u k-1 ). Applying this inductively, we see that p(w) ∈ Zp(w ). If w = w a k l , then using

we see that we can increase or decrease k depending on its sign until we get p(w) ∈ Zp(w ). We have shown that all the basis elements have the same image. Since x u is the image of the trivial representation, its image is 0 and the proof is complete.

We can now apply the same strategy as before : we restrict the resolution to H q and compute the corresponding six-term exact sequence. This yields Proposition 4.8. Let G = F n . Then, K 0 (C(H q )) = Z ⊕ Z 2 and K 1 (C(H q )) = Z n+1 .