Aït Khalil

Saïd

Youssef Souak
email: ysouak@inseec.com

Warda El Fardi
email: warda.el-fardi@u-bordeaux.fr

Is Talent Manager the New Job Title for Human Resources Manager?

Keywords: Talent manager, Human resources managers, rhetoric, career development JEL classification: J53, M12, M54

Talent management has become a central managerial concern, particularly in human resource management. Skillfully mixing HR and marketing functions, the talent manager is increasingly present in human resources job offers. However, "talent" as well as talent manager assignments are less clear and less precise. In this paper, we try to focus on the sense of "talent" given by talent managers themselves and on how they implement talent management in organization. Through a qualitative approach, we mobilize primary (semi-structured interviews) and secondary data (job offers) in a French context. The results show that organizations use "talent" as a rhetoric in order to improve their employer brand. The respondents indicate that talent management practices are so close to those of human resources management functions. What distinguishes the two functions the most is the proximity management.

Introduction

Talent management (TM) is a growing topic in human resource management (HRM) literature [START_REF] Thunnissen | A review of talent management:'infancy or adolescence?[END_REF]. How an organization identifies and manages talent will enable it to develop a competitive advantage and ensure its survival [START_REF] Schuler | Global talent management and global talent challenges: Strategic opportunities for IHRM[END_REF]. It was against the backdrop of a growing US economy and global competition for skilled employees that McKinsey consultants developed the concept of talent in the 1990s [START_REF] Michaels | The War for Talent[END_REF]. Talent is then considered as highly qualified populations or people with high potential [START_REF] Iles | Commentary on "The meaning of 'talent'in the world of work[END_REF].

TM helps optimize the human resources (HR) function by enabling the identification and management of employees with strategic human capital (d'Armagnac, Klarsfeld, & Martignon, 2016). TM is 'the process by which employers anticipate and meet their human capital needs' (Cappelli 2008, p. 1).

One of the issues that have emerged in the HRM literature is to define the concept and identify its evolution. It is clear that the subjectivity of this concept contributes to a wide variety of definitions and each author can give it the meaning he wants [START_REF] Ulrich | What is talent? Leader to leader[END_REF]. However, two conceptions of talent can be identified: subject and object.

The subject approach consists of focusing on the individuals who contribute to the organization's objectives and thinking about the practices that allow them to be differentiated. The object approach refers to talent as a set of knowledge and skills. This perspective is called inclusive because it addresses a more generalized part of the population (Daubnet-Siva et al., 2017). The literature is replete with work on TM practices and their organizational effects, but there is little work on the meaning of the concept [START_REF] Tansley | What do we mean by the term "talent" in talent management? Industrial and commercial training[END_REF]. This research focuses on how the word talent is understood from a managerial perspective. This research highlights the rhetorical aspects of the use of the word talent in organizational settings. While practitioners use careers as a TM tool, the academic literature does not study these elements together. From a theoretical point of view, this makes this field particularly interesting [START_REF] Mcdonnell | Talent management: a systematic review and future prospects[END_REF].

Furthermore, some studies have looked at the relationship between TM and HRM and show that the two fields can be frictional. Dries (2013) research shows, for example, that the boundaryless career model may be unsuitable for TM. This leads practitioners to question the possibilities of integrating the practices of the two fields. The development of TM practices within organizations justifies the importance of addressing them by studying TM concepts and practices and by using reflective evaluation [START_REF] Painter-Morland | Talent management: The good, the bad, and the possible[END_REF].

This research mobilizes the literature on talent and TM integrated with HRM. We investigate the missions of TM actors, the representations and practices articulated around talent, and their articulation with HR practices. In the first part, we present the literature on talent and TM in HRM by introducing a theoretical framework for understanding the missions of talent managers. In the second section, we present the methodology used and the research results.

Literature review

Talent: a variety of definitions and approaches

The literature does not provide a consistent definition of talent and TM [START_REF] Thunnissen | A review of talent management:'infancy or adolescence?[END_REF]. It may be understood as a rare set of rare competencies. TM is therefore defined as the detection, attraction, selection and recruitment, development, and retention of individuals who create value [START_REF] Ulrich | What is talent? Leader to leader[END_REF].

Two main approaches emerge in the literature: substantive and intersubjective [START_REF] Yeager | Mindsets That Promote Resilience: When Students Believe That Personal Characteristics Can Be Developed[END_REF]. The substantive approach echoes the meaning given to talent in antiquity, which represented an equivalent of gold used as currency (Picoche, 2006). Contemporary research emphasizes the links with this conception to explain exceptional physical performance and, and intellectual performance. Talent is then a set of natural and rare abilities that a minority of people have. This perspective is the most widespread in management science, both in terms of research and practice (Gallardo & Thunnissen, 2016). In view of these characteristics, talent can be a source of competitive advantage according to the resource-based theory (Barney, 2001).

In the intersubjective approach, talent is a dynamic object since it refers to individuals who interact continuously with their environment. This dynamic approach refutes the concept of accumulation of knowledge and skills in favor of the social dimension (Araújo & Davids, 2011). The role of practices that allow people to develop their own talents on the basis of exchanges with others is widely emphasized [START_REF] Swailes | Conceptualising inclusive talent management : Potential, possibilities and practicalities[END_REF]. Talent is then the ability to show agility, audacity, and intellectual and professional curiosity.

Another perspective is to look at talent through an inclusive/exclusive spectrum. The exclusive approach considers talents as high-performance or high-potential people [START_REF] Ulrich | What is talent? Leader to leader[END_REF]. It is also understood as having a strategic position in the hierarchy. Pivotal or critical talents are the central elements in the development of the organization. Talent can also be found outside the company through its ability to meet personnel needs [START_REF] Lewis | Talent management: A critical review[END_REF]. While the latter approach is more inclusive, the other approaches are considered more exclusive. They are also tools for making a typology of talents and the identification of talent-type-specific practices. The literature highlights the interest in these discussions through the implications for HRM (Luna-Arocas & Danvila-del-Valle, 2022). For example, exclusive TM practices and related communication can lead to a perception of unfairness by employees (Peterson et al, 2022). These phenomena are important because they can lead to counterproductive behavior by non-talented people. It emerges that one of the definitional issues is whether we consider talent as an object or a subject [START_REF] Thunnissen | A review of talent management:'infancy or adolescence?[END_REF].

The literature tends to define talent through the subject approach by designating them as 'competent incumbents' (Collings & Mellahi, 2009, p. 304), 'key employees' (Festing & Schäfer, 2014, p. 263), or 'employees who are considered talented' (Meyers & van Woerkom, 2014, p. 193). Other research views talent as personal attributes or scarce social resources [START_REF] Crane | Global talent management: A life cycle view of the interaction between human and social capital[END_REF].

The literature shows that associating talent with human capital is not incongruous [START_REF] Tansley | What do we mean by the term "talent" in talent management? Industrial and commercial training[END_REF]. The literature shows that talent and human capital must be studied jointly, and even interchangeably. Some authors agree that the 'object' approach of focusing on skills may be incomplete in capturing the reality of TM (Carne and Hartwell, 2019). This perspective neglects the core processes of selecting, integrating, developing, and retaining this human capital [START_REF] Farndale | The role of the corporate HR function in global talent management[END_REF]. While the concept of talent seems to be limited to the issue of human capital management, TM includes social dimensions such as socialization and leadership [START_REF] Dries | The psychology of talent management: A review and research agenda[END_REF].

The place of talent management in the human resources function

The literature mobilizes the word as a synonym for HR [START_REF] Iles | Commentary on "The meaning of 'talent'in the world of work[END_REF]. TM is then seen as an optimized tool for the HR function [START_REF] Lewis | Talent management: A critical review[END_REF]. In this framework, talents are considered as individuals with useful resources or high potential. Unlike the notion of talent, which refers to individual expertise in a key area, high potentials are those destined to occupy strategic positions in the hierarchy [START_REF] Ulrich | What is talent? Leader to leader[END_REF].

TM is one of many fields in HRM. Talent is an accumulation of human, political and social capital [START_REF] Farndale | The role of the corporate HR function in global talent management[END_REF]. This approach is largely based on the observation that resources that are useful to the firm are those that have the most value [START_REF] Pocztowski | Talent Management in Small and Medium Enterprises: Context, Practices and Outcomes[END_REF]. Another strand of the literature considers TM to be a shared function [START_REF] Lysenko | The new job position in human resources management the talent manager[END_REF]. It is a pivotal field in HRM and ensures its transformation. This new conception leads to the appearance of new functions and specific job titles. The emergence of this new field has consequently raised questions of alignment, but also a reorganization of missions within relational HRM models. Managers have seen their prerogatives reoriented toward TM practices [START_REF] Farndale | The role of the corporate HR function in global talent management[END_REF].

The literature does not highlight a univocal configuration of TM [START_REF] Lewis | Talent management: A critical review[END_REF]. Three trends emerge. The first stream proposes that HRM practices are universally TM, viewing talent as an accumulation of human capital. The second one views TM as a workforce management tool by contributing to the appropriate flow of people into the organization. The third-stream views TM as a meta-function of the organization, as these practices permeate all of its processes. It focuses on the inclusive perspective by identifying and developing talent and high potential, and on the inclusive perspective by underlying the importance of providing personal development for all employees. These elements lead to the observation that TM refers to the processes, cultural norms, and devices aimed at capturing and developing key individuals in view of the company's objectives. Some authors are critical of the interests behind TM. While some see it as a mere label for HRM to renew itself, others see it as a lever for optimizing HRM in order to propose exclusive practices and mobilize employees with particular potential (Ilies, 2013). The latter authors emphasize the desire to propose an individualized HRM based on the segmentation of the employed population. The work of [START_REF] Painter-Morland | Talent management: The good, the bad, and the possible[END_REF] shows that discussions on the articulation of HRM and TM are still relevant. They suggest that the complexity and size of the organization as well as the chosen perspective (individual or collective) could explain the intertwining or dissociation of the two fields. Moreover, these approaches are not static as they evolve over time and according to the context [START_REF] Thunnissen | A review of talent management:'infancy or adolescence?[END_REF]. Despite these conceptual differences, TM focuses on individuals who are identified as high performers, and possible current or future leaders [START_REF] Collings | Strategic talent management: A review and research agenda[END_REF][START_REF] Lewis | Talent management: A critical review[END_REF]. TM is therefore about acquiring, developing, and retaining these target individuals. When resources are not available internally, practices can look externally, relying on the employer branding, for example. This restrictive and monolithic approach, however, has been decried and appears limited in explaining the organizational reality of TM [START_REF] Stahl | Six principles of effective global talent management[END_REF]. For these reasons and because the literature is still embryonic, we propose to study actors' representations of talent and TM. Our work also focuses on the place of talent management in HRM.

Methodology

An inductive qualitative, interview-based methodology was used in line with the guidelines for qualitative methodology set out by [START_REF] Miles | Qualitative data analysis: An expanded sourcebook[END_REF]. The selection of respondents is based on a mixed-use of nonprobability sampling and snowball techniques. Initial participants were recruited first on LinkedIn and on the authors' professional networks.

We interviewed 13 French talent practitioners (talent managers, talent developers, talent acquisition specialists…) from different functional areas and industries (Consulting, Digital, Energy, …) to ensure a certain variety of understanding about how TM was practiced. Each interview lasts between 31 and 52 minutes. The respondents' mean tenure was 5 years. The average age is 35 years. Following the interview guide, we present an overview to the respondents about the research purpose emphasizing the confidentiality of their statements. The first questions concern the participants' professional career, the positions they held, and their tenure in the organization. This is followed by more specific questions regarding TM practices.

All interviews were recorded, transcribed, and loaded into Nvivo 12 software in order to perform content analysis. This latter was conducted manually and built on the anchored theory, as presented by [START_REF] Corbin | Basics of Qualitative Research[END_REF]. The coding process was performed iteratively as researchers ran back and forth between the transcripts and the evolving codes. The process is in three steps. Primarily, the purpose of the first-order analysis is to faithfully translate the Respondent's words in a way that makes no attempt to reduce the number of categories. Afterward, similarities existing among categories made it possible to decrease their number. Second-order themes were then linked to clustered dimensions. Codes have been set up using categories of conceptualization that make sense to the rubric.

Results and discussion

Drawing on 13 in-depth interviews, we investigated how TM is conceived and practiced by French organizations. Findings reveal that each employee is considered a talent by organizations. Being talented may not always be a technical ability, even if respondents did refer to employees with experience as well. 'As far as I'm concerned, it is almost as unique to the person as what they do best, it doesn't need to be a technical skill, it could be a problemsolving skill and a conflict resolution skill' (Respondent 4). Otherwise, talent should necessarily satisfy an organization's needs. 'Match talent to the requirements that will suit our goals' (Respondent 6).

Talent refers to various features such as technical expertise, social and personal qualities, learning willingness, and employer's values fit. 'We had a candidate who is technically very good, but his personality was not good enough to be part of the company' (Respondent 3). TM aims to create a proximate relationship with employees identified as talents. Consequently, organizations could attract and retain the best workers. Retaining them is expected to lead to a significant competitive advantage for the organization. 'At the heart of this is the willingness to drive value and commitment among employees in order to create value for the company' (Respondent 7). Furthermore, TM also serves to manage workforce plans. 'Organizations are changing, and we constantly need more resources, so we must be proactive. It is essential to get ahead of the curve' (Respondent 4).

Talent identification processes are focused on developing employees' potential, in line with the company's goal to promote career mobility, as well as focusing on more behavior-based considerations. 'The two classic criteria for identifying talent are organizational performance and developmental potential. This is our foremost talent reserve" (Respondent 9). For the organization, developing potential is about preparing the employee for executive roles. 'This is our foremost talent reserve' (Respondent 1).

When exclusive practices are implemented, though, the company makes no reference to it as such. "That the organization does not take a stand on talent identification lessens any frustration for those not identified as such" (Respondent 4). The existence of proprietary TM is not being explicitly communicated by talent managers, which makes it an organizational taboo. Rhetorical language appears even in inclusive TM approaches and there are examples of "talent acquisition specialists" or "talent developers" being used as "empty titles" (Respondent 10). Results indicate in fact talent practitioners are in charge of employees' career management and recruitment process.

The key elements of TM that emerged were training, skills development, and internal mobility. 'I strive to customize training courses with a high added value instead of general seminars' (Respondent 4). In practice, TM is intended to maximize the return on investment, both for the talent and more particularly for the organization.

As a result, mobility is often mentioned. It's about enabling talent to evolve within organizations. 'There are folks who have moved from consultant to project manager in less than a year' (Respondent 10). In addition, specific actions can be taken in TM practices: 'The same philosophy is applied to all employees, but in special cases, other approaches are needed to retain the best talent' (Respondent 12).

Concerning TM and the HR function relationship, the research findings highlight that the missions of talent managers encompass three main categories. Above all, it involves managing the whole recruitment process, starting with talent sourcing to onboarding; tasks include identifying talent ('school relationships', 'partnership building', 'employer brand', etc.).

Participants reported working under the direct control of the HR department, which may have similar assignments. In this situation, the HR department may have a sub-department responsible for TM. Talent managers operate in tandem with HR managers. 'When it comes to identifying and developing talent, we work closely with HR managers' (Respondent 11).

Talent managers' assignments seem alike to those of HR managers. 'It is all about the job title' (Respondent 4). The results agree that the usage of TM will differ whether an inclusive (object) or (subject) exclusive approach is adopted. Respondents insisted on proximity management as a distinguishable feature of TM and career management. 'Talent manager roles have evolved into something narrower than what used to exist, which was called career management' (Respondent 13). More emphasis is placed on accompanying specific individuals in managing and developing their careers.

Our results highlight the rhetorical use of talent by practitioners. They use it to refer to the entire internal or external pool of talent that can meet the organization's challenges. TM seems to be an integrated HRM area. Its creation is motivated by the goal of optimizing the HR function by providing improved career management systems.

Talent in practice is defined in many ways. A talented employee possesses both technical and non-technical competencies and is able to fulfill a particular organizational need. Once the talent has been identified, the organization ceases to use the 'talent' term among employees.

A major concern in the literature is whether or not to communicate talent identification. Indeed, as the talent pool is made known to all, it is possible that those excluded from selection may lose motivation. Likewise, without communicating the identification, frustration may also arise among top performers who do not feel sufficiently recognized as a talent [START_REF] Björkman | Talent or not? Employee reactions to talent identification[END_REF]. Thus, and in accordance with social exchange theory [START_REF] Blau | Exchange and power in social life[END_REF], they will adapt their behavior to how they believe the organization considers them competent and reliable While some organizations use inclusive and equality discourse, they nevertheless adopt exclusive talent identification practices that de facto exclude an entire segment of the workforce (Daubnet-Siva et al., 2017). What could happen is that some employees are confused about the meaning of the term 'talent' and therefore the identification of talent becomes more complex [START_REF] Sonnenberg | The role of talent-perception incongruence in effective talent management[END_REF]. Organizations adopt informal and hidden forms of TM by developing discrete support programs. We found that these approaches are inherently inclusive. Indeed, organizations open up the door for all employees to be 'talent', while building privileged relationships and individualized career development plans for those employees identified as essential.

Talent practitioners seem to be under the responsibility of the HR department. Still, most of the respondents stressed the shift from career management to TM. Ventolino & Mercier (2012) pointed out that the career manager's tasks are currently changing, due to internal and external constraints. Thus, for these researchers, TM should be more strategic in terms of managing talent careers. Specifically, what our study demonstrates is that talent management's presence is a clear message to both employees and candidates alike that employers are considering their aspirations. At the same time, these findings illustrate the difficulty talent managers face in dealing with these evolving expectations.

How Thunissien et al.'s (2013) view of TM as a top-down artifact to implement corporate mindset, corroborate our findings. As it fully or partially ignores the individuals' real insight, it fails to accurately assess their real agenda and adversely impacts their self-concept. It seems that using the term 'talent' to recruit, engage or activate HR appears to be unproblematic insofar as it reflects a good and valuable image for the target people. But then this mental connection tends to make it harder for managers to anticipate or detect potential deception or deception-related repercussions.

Conclusion

Whether in literature or in practice, the term talent is a notion that gives rise to many uses and understandings, sometimes giving the impression that the field remains poorly mastered. The purpose of this research was to shed light on practitioners' understanding and practices of the term.

TM is an important part of HRM because of its propensity to optimize operations. The authors who have been interested in this field have invested a lot of effort in trying to propose a stable definition. These discussions seem to lead to the observation that practices are based on an 'object' approach that considers talent as an accumulation of capital useful to the company, and a 'subject' approach that places at the center of the reflection the individuals holding the target competencies or likely to do so. In this research, we have highlighted the rhetorical uses of the term talent and its consequences on the managerial level.

The literature offers a discussion of the place of TM within the organization and specifically in relation to HRM. This research finally shows that TM practices are fully integrated into the HR function. The rhetorical use of the term 'talent' is used to enhance the role of HR managers and to promote a positive image with both internal clients and candidates. In fact, it appears that practitioners face tension between enhancing the image of the organization and making it attractive, while creating the conditions for fairness and equity. These problems are indeed observed when the system of identification and valorization is perceived as unfair or when a gap exists between discourse and practice. This research shows that this balancing act manifests itself in the promotion of an inclusive vision of TM while maintaining close and privileged relationships with talent. This ambidexterity relies heavily on HR practitioners mobilizing an informal structure. Finally, this research shows the importance of a concerted effort among the company's stakeholders to propose a commonly accepted definition of talent and to foster the perception of an inclusive approach by employees. This research has some limitations that should be made clear. In terms of the scope of the results, it should be noted that the level of generality is limited by the fact that we mobilized a limited number of interviews. This research highlights the role of rhetoric in gaining acceptance for TM practices while maintaining fairness and perceived equity. It would be interesting to study contexts where this ambivalence is detected by employees and to study the consequences it may have on their attitudes and behaviors. This research mobilizes only the perspective of managers, who are responsible for TM. Identifying the views of the populations to whom these practices and communications are directed would help to identify their effectiveness. Finally, this research essentially mobilizes interviews with managers and is therefore based on the discourse of these respondents. It would be useful to crossreference this data with institutional information or direct observations in order to study more deeply the elements of rhetoric.

Table 1 :

 1 Table 1 presents the demographic information of the sample. Demographic information of the sample

	Identification	Job title	Age	Gender	Tenure	Sector
	Respondent 1	Talent Manager 36	Female	3 years	Digital
	Respondent 2	Talent	34	Female	7 years	IT
		acquisition
		specialist
	Respondent 3	Talent Manager 32	Female	1 year	Digital
	Respondent 4	Talent Manager 36	Female	5 years	Digital
	Respondent 5	Talent Manager 33	Female	4 years	Consulting
	Respondent 6	Talent	37	Male	4 years	Energy
		acquisition
		recruiter
	Respondent 7	Project talent	43	Male	3 years	Energy
		management
	Respondent 8	Talent Manager 38	Male	3 years	Energy
	Respondent 9	Talent	33	Male	4 years	Telecommunicat
		acquisition				ions services
		specialist
	Respondent 10	Senior talent	37	Male	3 years	Textiles,
		manager				Clothing
	Respondent 11	Talent	34	Male	3 years	Media; culture;
		Developer				graphical
	Respondent 12	Talent	45	Female	5 years	IT
		acquisition
		recruiter


Vincent Despré

Camille Lanuel

Monique Teillaud

Computing an ε-net of a closed hyperbolic surface

An ε-net of a metric space X is a set of points P of X such that the balls of radius ε centered at points of P cover X, and two distinct points of P are at least ε apart. We present an algorithm to compute an ε-net of a closed hyperbolic surface and analyze its complexity.

Introduction

This paper focuses on hyperbolic surfaces, i.e., surfaces with a metric of constant negative curvature. These surfaces have been extensively examined from a mathematical perspective, due to their generic nature: any Riemannian surface of genus at least two can be conformally mapped to a unique hyperbolic surface [START_REF] Farkas | Riemann Surfaces[END_REF]Section IV.8].

Hyperbolic geometry also plays a key role in computer science. One of the most famous examples is found in the analysis of rotation distance in binary trees [START_REF] Daniel | Rotation distance, triangulations, and hyperbolic geometry[END_REF]. Hyperbolic geometry naturally emerges as a valuable tool for graph representation [START_REF] Eppstein | Squarepants in a tree: Sum of subtree clustering and hyperbolic pants decomposition[END_REF][START_REF] Eppstein | Limitations on realistic hyperbolic graph drawing[END_REF]. The hyperbolic plane also serves as the preferred model for illustrating the universal cover of surfaces with genus at least 2, which has proven to be crucial in the proof of purely topological results [START_REF] Despré | Computing the geometric intersection number of curves[END_REF][START_REF] Éric Colin De Verdière | Untangling graphs on surfaces[END_REF].

Delaunay triangulations of hyperbolic spaces and surfaces have been studied in the computational geometry community [START_REF] Bogdanov | Hyperbolic Delaunay complexes and Voronoi diagrams made practical[END_REF][START_REF] Iordanov | Implementing Delaunay triangulations of the Bolza surface[END_REF][START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF][START_REF] Ebbens | Minimal Delaunay triangulations of hyperbolic surfaces[END_REF]. In this line, we adapt Shewchuk's Delaunay refinement algorithm [START_REF] Richard | Delaunay refinement algorithms for triangular mesh generation[END_REF] to construct ε-nets of hyperbolic surfaces, opening the door to the design of efficient approximation algorithms. To the best of our knowledge, this is the first result of this kind.

Let us recall definitions [START_REF] Kenneth | Building triangulations using ϵ-nets[END_REF]. Let (X, d) be a metric space and ε > 0. A set P ⊂ X is an ε-covering if ∀x ∈ X, d(x, P ) ⩽ ε, i.e., if the closed balls of radius ε centered at each p ∈ P cover X. It is an ε-packing if ∀p ̸ = q ∈ P, d(p, q) ⩾ ε, i.e., if the open balls of radius ε/2 centered at each p ∈ P are pairwise disjoint. An ε-net is both an ε-covering and an ε-packing. In this paper, we prove: Proposition 1. Any ε-net of a closed hyperbolic surface S of genus g and systole

σ contains N ⩽ 16(g -1) 1/ε 2 + 1/σ 2 points. If ε < σ, then N ⩽ 16(g -1)/ε 2 .
The case when ε < σ corresponds to the situation when the surface has no ε-thin part (see Section 2.2). For a fixed surface, the complexity is then O(1/ε 4 ).

Proposition 2. The Delaunay refinement algorithm computes an ε-net using at most 10 + C

′ h Diam(S) 6g-4 N 2 + (N -1)(144g
The first result can be regarded as folklore. We prove it in Section 3 for completeness. The second proposition rises interesting obstacles to deal with. In particular, Shewchuk's refinement adds circumcenters of some triangles, which is not straightforward in our context, as locating a new point requires to construct a portion of the universal cover of the surface. We manage to bound the size of this portion.

Background on hyperbolic surfaces and notation

We refer the reader to textbooks for more details, e.g. [START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF][START_REF] Beardon | The Geometry of Discrete Groups[END_REF].

A closed hyperbolic surface can be seen as the quotient H 2 /Γ of the hyperbolic plane H 2 under the action of a group Γ of orientation-preserving isometries. Throughout the paper, objects in H 2 are denoted with a tilde •, while objects on S are denoted without. In particular, for an object o on S, o denotes any of its lifts in H 2 . To simplify the language, we often use the term copy to refer to an image of an object in H 2 by an element of Γ.

We work with the Poincaré disk model in which the hyperbolic plane H 2 is represented as the unit disk of the complex plane C. The unit circle consists of points at infinity. The geodesics are either diameters of the unit disk, or circular arcs that meet the boundary circle orthogonally. The hyperbolic circles are Euclidean circles (but their hyperbolic and Euclidean centers differ). Orientationpreserving isometries are represented as matrices in C 2×2 .

Delaunay triangulation and Dirichlet domain

A triangulation T of S is a partition of S into triangles; note that edges may be loops. A triangulation of S is a Delaunay triangulation if for each triangle t of T and any of its lifts t in H 2 , the open disk circumscribing t contains no vertex of the (infinite) lift of T in H 2 [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]. The Voronoi diagram is the dual of the Delaunay triangulation. The Dirichlet domain D x of a point x ∈ H 2 is the (closed) cell of x in the Voronoi diagram of its (infinite) orbit Γ x. Unlike the Euclidean case, Γ is non-commutative, and the combinatorics of a Dirichlet domain depends on the point x (Figure 1). The number k of sides of D x satisfies 4g ⩽ k ⩽ 12g -6 (see, e.g., [START_REF] Despré | Computing a Dirichlet domain for a hyperbolic surface[END_REF]).

In this paper, we assume that the input surface S is given by a Delaunay triangulation having a single vertex b, i.e., all Delaunay edges are loops based in b. The point b is arbitrary. This introduces no restriction, as such a representation can be computed for any closed hyperbolic surface, starting from a standard representation by a fundamental domain and side pairings [START_REF] Despré | Computing a Dirichlet domain for a hyperbolic surface[END_REF]. as s i , i = 0, . . . , k -1 and the corresponding side pairings as γ i , i = 0, . . . , k -1 (here, side pairings are pairwise inverses).

Thin and thick parts

The injectivity radius r x (S) of S at a point x is the supremum of all r > 0 such that the open ball of radius r centered at x, B(x, r) = {y ∈ S | δ S (x, y) < r}, where δ S is the distance on S, is isometric to a disk in H 2 . In particular, B(x, r) is a topologically embedded disk on S for all r ⩽ r x (S). The systole σ of a surface is the length of its shortest non-contractible curve, which we also denote by σ. The systole is related to the injectivity radius:

σ = 2 • inf {r x (S) | x ∈ S}.
For any ε > 0, the ε-thin

part of S is S t ε = {x ∈ S | r x (S) ⩽ ε/2}, and its ε-thick part is S T ε = S \ S t ε . Observe that if ε < σ, then there is no ε-thin part. σ ⩽ ε

Proof of proposition 1

Let P be an ε-packing of S. The open balls of radius ε/2 centered at the points of P on the ε-thick part S T ε are isometric to disks in H 2 and are pairwise disjoint. The area of such a disk centered at a point

p is A B p, ε 2 = 4π sinh 2 ε 4 [1, Theorem 7.2.2]. Since sinh(x) ⩾ x for all x ⩾ 0, we have A B p, ε 2 ⩾ πε 2 /4.
Let N T be the number of points of P on the ε-thick part S T ε . By the Gauss-Bonnet theorem, the area of the surface S is A(S) = 4π(g -1). Summing the above inequality over all the points in P ∩ S T ε , we obtain

N T πε 2 /4 ⩽ p∈P ∩S T ε A B p, ε 2 ⩽ 4π(g -1)
, thus

N T ⩽ 16(g -1) ε 2 . ( 1
)
The open balls of radius ε/2 in the ε-thin part S t ε , if it exists, that is if σ ⩽ ε, are also pairwise disjoint, but they are not isometric to disks in H 2 . However, by definition, the open balls of radius σ/2 are isometric to disks in H 2 . We can apply the reasoning that led to inequality (1) for σ instead of ε, and obtain a bound on the number of points of P on the thin part S t ε : N t ⩽ 16(g -1)/σ 2 . The bound on the total number of points of P follows.

Our algorithm is inspired by Shewchuk's Delaunay refinement [START_REF] Richard | Delaunay refinement algorithms for triangular mesh generation[END_REF]. The general idea is to break each Delaunay triangle whose circumcircle has a radius greater than ε by inserting its circumscribing center in the triangulation.

We reuse the data structure proposed by Despré et al. for computing the Delaunay triangulation of a surface by edge flips [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]. A triangulation of S is represented by

• its vertices: a vertex p has constant-time access to its lift p b in D b and one of its incident triangles;

• and its triangles: a triangle ∆ has constant-time access to its three vertices p ∆ 0 , p ∆ 1 , p ∆ 2 , its three adjacent triangles, and three isometries

γ ∆ 0 = 1 Γ , γ ∆ 1 , γ ∆ 2 in Γ defined as follows. A triangle ∆ = (p ∆ 0 ; p ∆ 1 ; p ∆ 2 )
does not always have a lift entirely included in D b . However, it always has at least one lift with at least one vertex in D b (see Figure 3). Let us choose such a lift and denote it as ∆ 0 ; up to a re-indexing of its vertices, p ∆ 0 ∈ D b . Then γ ∆ 1 and γ ∆ 2 are the isometries such that the other two vertices of ∆ 0 are γ ∆ 1 p ∆ 1 and γ ∆ 2 p ∆ 2 . Note that the other lifts of ∆ having at least one vertex in D b can be retrieved by applying the inverses of these isometries to ∆ 0 . The union, on all triangles of the triangulation of S, of their lifts with at least one vertex in D b covers the fundamental domain D b . We denote as DT (•) the Delaunay triangulation of a set of points on S.

D b ∆ 0 p ∆ 0 p ∆ 2 p ∆ 1 γ ∆ 1 γ ∆ 2 γ ∆ 0 = 1 Γ
Let us fix ε > 0. In a first step, the set of points is initialized as P 1 = {b}.

At each step i ⩾ 2, the algorithm inserts the circumscribing center c of a triangle ∆ ε whose radius is greater than ε. The set of points is updated as P i = P i-1 ∪{c} as well as the Delaunay triangulation DT (P i ). To do so, several operations are needed.

We first compute the radius of ∆ 0 for every triangle ∆ of DT (P i-1 ), until a triangle ∆ ε whose radius is at least ε is found. 2 The circumcenter c of the lift ∆ ε 0 is a lift of c, but it does not necessarily lie in D b . This can be checked by testing whether b and c lie on the same side of the supporting line of each side of D b . The Delaunay triangulation DT (P i ) of P i = P i-1 ∪ {c} can then be computed. First, the triangle ∆ c of DT (P i-1 ) containing c is found by naively checking if c b lies in one of the (at most three) lifts of each triangle ∆ in DT (P i-1 ) having a vertex in D b . This can be done by testing, for each edge, whether c b and the third vertex of the triangle lie on the same side of its supporting line. Then ∆ c is split into three by creating an edge between c and its three vertices. In the data structure, the three isometries stored in each new triangle are 1 Γ for c, and the corresponding isometries in ∆ c for the other two vertices. Then DT (P i ) is computed with a sequence of flips and the data structure is updated [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF].

To actually insert c into DT (P

The termination of the algorithm is quite obvious. At step i = 1, the ε-packing P 1 consists of one point. At each step i ⩾ 2, the point added to P i is the circumcenter of a Delaunay triangle whose radius is at least ε. Because no vertex lies in the interior of a Delaunay disk, the center added is at distance at least ε from any point of P i . By induction, P i is an ε-packing containing i points. By Proposition 1, the algorithm must terminate after a finite number N -1 of insertions. It returns an ε-packing P N of cardinality N .

It remains to show that P N is an ε-covering of S. Let x be a point on S. It lies in a triangle ∆ of DT (P N ). Let ∆ be a lift of ∆ and x the lift of x lying in ∆.

The circumdisk of ∆ has a radius r ⩽ ε. There is a vertex of ∆ whose distance to x is at most r (see Lemma 2 in appendix). That vertex is a lift of a point of P N by definition of ∆. It follows that δ S (x, P N ) ⩽ ε, therefore P N is an ε-net. This establishes the first claim of Proposition 2.

Algorithm analysis

This section is devoted to proving the complexity announced in Proposition 2.

The following operations take O(1) time in the real RAM model and we consider them as elementary operations:

• Computing ∆ 0 from a triangle ∆ of the data structure (see Section 4 for notation);

• Computing the radius or the center of the circumcircle of a triangle in H 2 ;

• Deciding if a point lies on the right or the left side of an oriented geodesic segment in H 2 ;

• Flipping an edge of a triangulation [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]Section 4.1].

At the beginning of a step i ⩾ 2, P i-1 contains i-1 points, the Euler characteristic shows that DT (P i-1 ) has 2i + 4g -2 triangles, which gives the cost of finding ∆ ε .

Recall that the number k of sides of D b is at most 12g -6 (see Section 2). Determining whether c lies in (a given copy of) D b thus requires at most 12g -6 elementary operations. The algorithm tests the copies of D b that intersect the geodesic segment p ∆ ε 0 c. Since ∆ ε 0 is a triangle of DT P i-1 , its circumcircle does not contain any other lift of p ∆ ε 0 , so p ∆ ε 0 is the closest lift of p ∆ ε 0 to c. The geodesic segment p ∆ ε 0 c is thus a lift of a distance path 4 on S, what is called a distance path in H 2 . By [9, Proposition 14], every side of D b is either a distance path, or the concatenation of two distance paths. As two distance paths that do not have a subarc in common, which is the case here, can intersect at most once [9, Finding ∆ c in DT (P i-1 ) when c b is known requires at most 9(2i + 4g -2) elementary operations since it amounts to checking the three edges of at most three lifts of each triangle. The update of the data structure when splitting the triangle containing c into three is done in 8 elementary operations (deleting the triangle that contains c, adding c to the list of vertices, creating 3 triangles and 3 isometries).

Adding the above costs for step i, locating c in DT (P i-1 ) and splitting the triangle containing it costs at most 10(2i + 4g -2) + (12g -6) 2 + 9 elementary operations.

The flips are counted globally for all steps, which concludes the proof of Proposition 2.

Lemma 1. The total number of flipped edges during the execution of the algorithm is at most C ′

h Diam(S) 6g-4 N 2 , where C ′ h is a constant depending on the metric h of S, and Diam(S) is the diameter of S.

The proof of this lemma mimicks the proofs in [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]. The situation is quite different here, as the points are inserted incrementally and the flips are done at each insertion, whereas all points are know in advance in [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF], which requires to rewrite a complete proof.

Proof. Denote as T 1 = DT (P 1 ), T 2 , . . . , T K the sequence of triangulations appearing during our algorithm. For j ⩾ 1, T j+1 is obtained from T j either by flipping an edge, or by splitting a triangle into three from a new vertex. Every triangulation is geometric: it is equivalent to a triangulation whose edges are geodesic segments that do not intersect in their interior. Flipping an edge maintains the property [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]; splitting a triangle clearly maintains it, too.

We associate to any triangulation T of S a polyhedral surface Σ in R 3 as in [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]Section 2.3]: the vertices of Σ are obtained from the vertices of the (infinite) lift T of T by the stereographic projection onto the unit sphere S 2 . The (infinite) surface Σ is convex if and only if T is a Delaunay triangulation of S. Let us show that Σ j+1 contains Σ j for each j ⩾ 1, i.e., Σ j+1 lies between Σ j and S 2 . The case when T j+1 is obtained by flipping a non-Delaunay edge e of T j is studied in [11, Section 2.3]: Σ j is concave at each edge projected from a lift of e, and after the flip Σ j+1 is convex at all the new edges. Let us now examine the case when T j+1 is obtained by splitting a triangle of T j into three from a new vertex. In this case, T j is a Delaunay triangulation, so Σ j is convex, but the edges left in Σ j+1 from each triangle of T j that is split from the new vertex on S 2 are generally not convex. The surface Σ j is thus contained in Σ j+1 as well. As a result, every Σ j ′ with j ′ > j contains Σ j . If an edge is flipped at a step i of the algorithm, the corresponding line segment becomes interior to the polyhedral surface and all the following, and it can never reappear.

We can now observe that no flip will ever create an edge longer than 8 Diam(S). Consider the fundamental domain Ω b consisting of one lift of each triangle of T 1 incident to b. For all x ∈ Ω b , δ H 2 ( x, b) < 2 Diam(S), where δ H 2 is the distance in H 2 . This is because x belongs to the circumdisk of the triangle t ∈ T 1 it lies in, and b lies on its boundary. That circumdisk must have a radius r < Diam(S), otherwise it would contain at least one lift of every point of S. In particular, it would contain a lift of a vertex of t in its interior, which is impossible since t is a Delaunay triangle. Let e be an edge created by a flip and let v be the lift of its midpoint v lying in Ω b . The fundamental domain Ω b is strictly included in the disk of radius 4 Diam(S) and centered at v: indeed, if x ∈ Ω b then δ H 2 ( x, v) ⩽ δ H 2 ( x, b) + δ H 2 ( b, v) < 4 Diam(S). The proof of [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]Lemma 10], replacing 2∆(T ) by 8 Diam(S) and Ω with Ω b , shows that e cannot be longer than 8 Diam(S).

The proof of [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]Theorem 19], replacing 2∆(T ) with 8 Diam(S), proves that a Delaunay flip algorithm performed on a triangulation of S with n vertices flips at most C ′ h Diam(S) 6g-4 n 2 edges, where C ′ h is a constant5 depending on the metric h of S. Since the edges that have been flipped cannot reappear, the number f i of flipped edges at step i of our algorithm satisfies f i ⩽ C ′ h Diam(S) 6g-4 i 2 -i-1 j=2 f j .

It follows that i j=2 f j ⩽ C ′ h Diam(S) 6g-4 i 2 for each i = 2, . . . , N .

Note that the bound comes from the best upper bound O(Diam(S) 6g-4 ) known so far for the flip algorithm [START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF]. The actual complexity of the flip algorithm may be much better [START_REF] Despré | Experimental analysis of Delaunay flip algorithms on genus two hyperbolic surfaces[END_REF].

bisector between x and c, separates x from ∆, so x cannot be in ∆ (Figure 4, left). Thus, at least one vertex of ∆ must lie on C ∆ ∩ D x , that is, at distance at most r from x.

1 Figure 1 :

 11 Figure 1: Dirichlet domains for the Bolza surface (g = 2). The domain on the left has 4g = 8 sides and the one on the right has 12g -6 = 18 sides. Figure from [3].

Figure 2 :

 2 Figure 2: Thick and thin (red) parts of a hyperbolic surface. Disks of radius ε are shown in blue.

Figure 3 :

 3 Figure 3: Example of a triangle ∆ having three lifts with one vertex in D b (the hyperbolic triangles are schematically represented with straight edges).

  i-1 ), we need to find the lift c b of c that lies in D b . If c lies in D b , then c b = c. Otherwise, the algorithm walks in the tiling {γD b } γ∈Γ of H 2 along the geodesic segment p ∆ ε 0 c. The first copy of D b traversed by p ∆ ε 0 c is found by looking for the side s j1 , j 1 ∈ {0, . . . , k -1} of D b intersecting it. 3 The walk along p ∆ ε 0 c continues in γ j1 D b , and so on, until the copy γ jn . . . γ j1 D b containing c is found. Then c b = γ -1 j1 . . . γ -1 jn c. Note that the walk still works when p ∆ ε 0 c goes through a vertex of a copy of D b .

Lemma 8 ]

 8 , p ∆ ε 0 c traverses at most 2k sides of copies of D b . If an intersection occurs at a vertex of degree d of a copy of D b , then this counts for d intersections. Searching the copy of D b containing c hence requires k 2 ⩽ (12g -6) 2 elementary operations. Computing c b costs 1 operation.

Figure 4 :

 4 Figure 4: Illustration of the proof of Lemma 2 in Euclidean geometry (for simplicity). The points c and x belong in each other's disk of radius r. Right: If no vertex of ∆ lie on C ∆ ∩ D x (bold), x is separated from ∆ by the dashed geodesic.

The common basepoint is denoted as b ′′ in[START_REF] Despré | Computing a Dirichlet domain for a hyperbolic surface[END_REF].

Construction of the ε-netThe input of the algorithm consists of the Delaunay triangulation of S with a single vertex b, together with the Dirichlet domain D b of a lift b and the group Γ generated by side-pairings. As mentioned in Section 2.1, this does not induce any loss of generality.

Of course a priority queue could be used to improve the complexity of this search. We accept a linear complexity for simplicity, as this is not the dominant operation in the algorithm.

To check whether two geodesic segments x 1 x 2 and y 1 y 2 intersect, we check whether x 1 and x 2 lie on opposite sides of the supporting line of y 1 y 2 , and we run the same test, swapping the roles of x and y.

A distance path on S is a shortest path between two points. It is necessarily a geodesic segment, but not all geodesic segments are distance paths since they only locally minimize distances.

Our constant C ′ h equals 8C h , where C h is the constant in[START_REF] Despré | Flipping geometric triangulations on hyperbolic surfaces[END_REF].

Acknowledgements. The authors wish to thank Hugo Parlier for interesting discussions.

A Lemma for the proof of correctness (Section 4)

Lemma 2. Let ∆ be a triangle of H 2 and x ∈ ∆. Denote r as the radius of the circumcircle of ∆. Then there exists a vertex of ∆ whose distance to x is at most r. This lemma holds in both the Euclidean and the hyperbolic planes, as the following proof uses arguments that function in both settings.

Proof. Let D x be the closed disk of radius r centered in x, D ∆ be the circumdisk of ∆, and c be its circumcenter. If x = c, the two disks are equal and the result is trivial.

Suppose that x ̸ = c. Denote C ∆ (resp. C x ) as the circle bounding the disk D ∆ (resp. D x ). The distance between x and c is at most r because x lies in ∆, so x ∈ D ∆ and c ∈ D x . Therefore, the two disks intersect and their intersection is not a singleton. Moreover, D ∆ ̸ = D x , so C ∆ and C x intersect exactly twice (Figure 4). The three vertices of ∆ lie on the circle C ∆ . But if their distance to x is greater than r, i.e. if they belong to C ∆ \ D x , then the geodesic passing through the two intersection points of the circles, that is, the perpendicular


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Gambierol blocks a K(+) current component affecting action potential duration in fetal adrenomedullary chromaffin cells. Is this action sufficient to modulate catecholamine release?

Jordi Molgó, Roland Bournaud, Sébastien Schlumberger, Makoto Sasaki,

Haruhiko Fuwa, Denis Servent, Evelyne Benoit

To cite this version:

jordi.molgo@cea.fr

The genera Gambierdiscus and Fukuyoa produce numerous ladder polycyclic ether compounds including gambierol, characterized by a transfused octacyclic polyether core, and the identified ciguatoxins, responsible for ciguatera poisoning. Gambierol inhibits voltage-gated K+ (Kv) channels in various cells, and motor nerve terminals and enhances acetylcholine release. The purpose of this work was to study the effects of gambierol on cultured rat fetal adrenomedullary chromaffin cells. These excitable cells have different types of Kv channels and release catecholamines. Perforated patch-clamp current-recordings revealed that gambierol (100 nM) only blocked a small component of the total outward K+ current, and did not affect calciumâactivated K+ channels (KCa) and ATP-sensitive K+ (KATP) channels. When KATP and KCa channel activation were blocked, the gambierol concentration inhibiting 50 % of the K+ current-component (IC50) was 7.6 ± 1.10 nM (n = 9). Gambierol also slowed the kinetics of K+ current activation. The recording of all-or-none action potentials in chromaffin cells disclosed that gambierol (50 nM) prolonged by 33 % their repolarizing phase. Gambierol did not modify the membrane potential following 15-second depolarizing current-steps, and did not increase the number of exocytotic catecholamine release events, as determined by simultaneous patchclamp and single-cell amperometry recordings. Under, the same conditions gambierol enhanced catecholamine secretion provided KCa and KATP channels were blocked. In conclusion, the specific inhibition of Kv channels by gambierol is not enough to modulate catecholaminesecretion, emphasizing the key role played by KCa and KATP channels.

Gambierol blocks a K+ current component affecting action potential duration in fetal adrenomedullary chromaffin cells. Is this action sufficient to modulate catecholamine release?

  Jordi Molgo 1 , Roland Bournaud 2 , Sébastien Schlumberger 2 , Makoto Sasaki 3 , Haruhiko Fuwa 4 , Denis Servent 5 , Evelyne Benoit 5 1 Université Paris-Saclay, CEA, INRAE, Institut des sciences du vivant Frédéric Joliot, Départem -Bâtiment 152, Gif sur Yvette, F-91191, France. 2 Laboratoire de Neurobiologie Cellulaire et Moléculaire, Gif-sur-Yvette, France. 3 Tohoku University, Graduate School of Life Sciences, Rm E516, 468-1, Aramaki Aza-Aoba, Aoba-ku Sendai 980-8572, Japan. 4 Chuo University, Faculty of Science and Engineering, Department of Applied Chemistry, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan. 5 Université Paris-Saclay, CEA, INRAE, Institut des sciences du vivant Frédéric Joliot, Département Médicaments et Technologies pour la Santé (DMTS), Service d'Ingénierie Moléculaire pour la Santé (SI).

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Jordi Molgó, Roland Bournaud, Sébastien Schlumberger, Makoto Sasaki, Haruhiko Fuwa, et al.. Gambierol blocks a K(+) current component affecting action potential duration in fetal adrenomedullary chromaffin cells. Is this action sufficient to modulate catecholamine release?. ICHA 2021 -19th International conference on Harmful algae, Oct 2021, La Paz (Virtual), Mexico. hal-04471073v2