- Combinatorial nonpositive curvature: CAT(0) cube complexes, median graphs, Helly graphs, weakly modular graphs, lattices, Garside categories...
- Coarse nonpositive curvature: Gromov-hyperbolic spaces, hierarchically hyperbolic spaces, coarse median spaces, coarsely Helly graphs
- Continuous nonpositive curvature: symmetric spaces of noncompact type, buildings, Teichmüller spaces, CAT(0) spaces, piecewise L^p spaces, injective metric spaces, piecewise polyhedral metric spaces, geodesic bicombings, Busemann-convexity, strong bolicity, CUB spaces...
- Braid groups, Coxeter groups, Artin-Tits groups, fundamental groups of arrangements of hyperplanes, complex braid groups, Garside groups...
- Mapping class groups of surfaces, hierarchically hyperbolic groups...
- Semi-simple Lie groups over local fields, lattices, rigidity
- Closed subgroups of locally compact topological groups
- Fundamental groups of surfaces, random walks and stationary probability measures

- "New Garside structures and applications to Artin groups", with Jingyin Huang, preprint (2023), arXiv.
- "Lp metrics on cell complexes", with Nima Hoda and Harry Petyt, preprint (2023), arXiv.
- "A link condition for simplicial complexes, and CUB spaces", preprint (2022), arXiv.
- "Locally elliptic actions, torsion groups, and nonpositively curved spaces", with Damian Osajda, preprint (2021), arXiv.
- "Tangent bundles of hyperbolic spaces and proper affine actions on Lp spaces", with Indira Chatterji, François Dahmani and Jean Lécureux, preprint (2019), arXiv.
- "Horofunction Compactifications of Symmetric Spaces", with Anna-Sofie Schilling, Anna Wienhard and Cormac Walsh, preprint (2018), arXiv.

Garside groups are combinatorial generalizations of braid groups which enjoy many nice algebraic, geometric, and algorithmic properties. In this article we propose a method for turning the direct product of a group G by ℤ into a Garside group, under simple assumptions on G. This method gives many new examples of Garside groups, including groups satisfying certain small cancellation condition (including surface groups) and groups with a systolic presentation. Our method also works for a large class of Artin groups, leading to many new group theoretic, geometric and topological consequences for them. In particular, we prove new cases of K(π,1)-conjecture for some hyperbolic type Artin groups.

Motivated by the observation that groups can be effectively studied using metric spaces modelled on l1, l2, and l∞ geometry, we consider cell complexes equipped with an lp metric for arbitrary p. Under weak conditions that can be checked locally, we establish nonpositive curvature properties of these complexes, such as Busemann- convexity and strong bolicity. We also provide detailed information on the geodesics of these metrics in the special case of CAT(0) cube complexes.

We motivate the study of metric spaces with a unique convex geodesic bicombing, which we call CUB spaces. These encompass many classical notions of nonpositive curvature, such as CAT(0) spaces and Busemann-convex spaces. Groups having a geometric action on a CUB space enjoy numerous properties.

We want to know when a simplicial complex, endowed with a natural polyhedral metric, is CUB. We establish a link condition, stating essentially that the complex is locally a lattice. This generalizes Gromov's link condition for cube complexes, for the ℓ∞ metric.

The link condition applies to numerous examples, including Euclidean buildings, simplices of groups, Artin complexes of Euclidean Artin groups, (weak) Garside groups, some arcs and curve complexes, and minimal spanning surfaces of knots.

Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic (that is, every element has a bounded orbit) actions by automorphisms of finitely generated groups on finite dimensional nonpositively curved complexes have global fixed points. In particular, finitely generated torsion groups cannot act without fixed points on such spaces.

We prove these conjectures for a wide class of complexes, including all infinite families of Euclidean buildings, Helly complexes, some graphical small cancellation and systolic complexes, uniformly locally finite Gromov hyperbolic graphs. We present numerous consequences of these result, e.g. concerning the automatic continuity.

On the way we prove several results concerning automorphisms of Helly graphs. They are of independent interest and include a classification result: any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic, with rational translation length. One consequence is that groups with distorted elements cannot act properly on such graphs. We also present and study a new notion of geodesic clique paths. Their local-to-global properties are crucial in our proof of ellipticity results.

We define the notion of a negatively curved tangent bundle of a metric measured space. We prove that, when a group G acts on a metric measured space X with a negatively curved tangent bundle, then G acts on some Lp space, and that this action is proper under suitable assumptions. We then check that this result applies to the case when X is Gromov-hyperbolic.

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.

- "Lattices, injective metrics and the K(π,1) conjecture", accepted to Algebraic & Geometric Topology (2023), arXiv.
- "Automorphisms and subdivisions of Helly graphs", accepted to J. Topology & Analysis arXiv (2023).
- "Lattices, Garside structures and weakly modular graphs", with Jingyin Huang, accepted to Journal of Algebra (2023), arXiv.
- "Random walk speed is a proper function on Teichmüller space", with Aitor Azemar , Vaibhav Gadre, Sébastien Gouëzel, Pablo Lessa and Caglar Uyanik, to appear in Journal of Modern Dynamics, Vol.19 (2023), arXiv.
- "Injective metrics on buildings and symmetric spaces", Bull. LMS, Vol. 54, Issue 6, 2297-2313 (2022).
- "Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity", with Nima Hoda and Harry Petyt, accepted to Geometry & Topology (2021), arXiv.
- "XXL type Artin groups are CAT(0) and acylindrically hyperbolic", Annales de l'Institut Fourier, Tome 72, no. 6, 2541-2555 (2022).
- "Cubulation of some triangle-free Artin-Tits groups", Groups, Geometry and Dynamics, Vol. 16, No. 1, 287-304 (2022), arXiv.
- "Virtually cocompactly cubulated Artin-Tits groups", IMRN (2021), 4, 2919-2961, arXiv.
- "Hyperbolic rigidity of higher rank lattices", with an appendix by Camille Horbez and Vincent Guirardel, Ann. Sci ENS (2020), 4e série, t.53, 437-468, arXiv.
- "Higher rank lattices are not coarse median", Algebraic and Geometric Topology 16 (2016), no. 5.
- "The 6-strand braid group is CAT(0)", with Dawid Kielak and Petra Schwer, Geometriae Dedicata 182 (2016), no. 2.
- "Visual limits of flats in symmetric spaces and Euclidean buildings", published in Transformation Groups 18, Issue 4 (2013), 1055-1089.
- "Compactification de Thurston d'espaces de réseaux marqués et de l'espace de Torelli", published in Groups, Geometry and Dynamics, Vol.9, Issue 2 (2015), 331-368.
- "Compactification de Chabauty de l'espace des sous-groupes de Cartan de SL_n(R)", published in Mathematische Zeitschrift 274, Issue 1 (2013), 573-601.
- "Compactification de Chabauty des espaces symétriques de type non compact", published in Journal of Lie Theory 20 (2010), 437-468.
- "L'espace des sous-groupes fermés du groupe R x Z", published in Algebraic and Geometric Topology 10 (2010), 1395-1415.

Starting with a lattice with an action of Z or R, we build a Helly graph or an injective metric space. We deduce that the sup orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces.

We apply this to show that any Garside group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise sup metric on any Euclidean building of type A extended, B, C or D is injective, and its thickening is a Helly graph.

Concerning Artin groups of Euclidean types A and C, we show that the natural piecewise sup metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the K(π,1) conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.

We study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graphs, in terms of cliques which are intersections of balls. We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.

In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural An tilde shaped simplicies, including Euclidean buildings of type An tilde and Cayley graphs of Garside groups and their quotients by the Garside elements.

All these examples fit into the more general setting of lattices with order-increasing ℤ-actions and the associated lattice quotients proposed in a previous work by the first named author. We show that both the lattice quotients and the lattices themselves give rise to weakly modular graphs, which is a form of combinatorial non-positive curvature.

We also show that several other complexes fit into this setting of lattices/lattice quotients, hence our result applies, including Artin complexes of Artin-Tits groups of type An tilde, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Along the way, we also clarify the relationship between categorical Garside structure, lattices with ℤ action and different classes of complexes studied this article. We use this point of view to describe the first examples of Garside groups with exotic properties, like non-linearity or rigidity results.

Consider a closed surface M with negative Euler characteristic, and an admissible probability measure on the fundamental group of M with finite first moment. Corresponding to each point in the Teichmüller space of M, there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichmüller space of M, and we relate the growth of the speed to the Teichmüller distance to a basepoint. One key argument is an adaptation of Gou\"ezel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.

In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls.

On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise ℓ∞ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local
fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices.

On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed
with a natural invariant Finsler metric, restricting to an ℓ∞ norm on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on
injective metric spaces. We identify the injective hull of the symmetric space of GL(n,R) as the space of all norms on
R^n. The only exception is the special linear group: if n=3 or n is at least 5 and K is a local field, we show that
SL(n,K) does not act properly and coboundedly on an injective metric space.

We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that any hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that any coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups---including mapping class groups of surfaces---are coarsely injective and coarsely injective groups are strongly shortcut. Using these results we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, are of type FP∞, have finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.

We describe a simple locally CAT(0) classifying space for extra extra large type Artin groups (with all labels at least 5). Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that extra extra large type Artin groups are acylindrically hyperbolic. Together with Property RD proved by Ciabonu, Holt and Rees, the CAT(0) property implies the Baum-Connes conjecture for all extra extra large type Artin groups.

We prove that some classes of triangle-free Artin-Tits groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin-Tits groups that are properly cubulated but cannot be cocompactly cubulated, even virtually. The existence of such a proper action has many interesting consequences for the group, notably the Haagerup property, and the Baum-Connes conjecture with coefficients.

We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-dimensional type. A particular case is that for n at least 4, the n-strand braid group is not virtually cocompactly cubulated.

We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem nor on bounded cohomology. More generally, we prove that any morphism from a higher rank lattice to a hierarchically hyperbolic group has finite image. In the Appendix, Vincent Guirardel and Camille Horbez deduce rigidity results for morphisms from a higher rank lattice to various outer automorphism groups.

We show that symmetric spaces and thick affine buildings which are not of spherical type A_1^r have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median.

We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.

Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compactification of the space of maximal flats of X. We then completely determine the possible degenerations of apartments when X is of rank 1, associated to a classical group of rank 2 or to PGL(4). In particular, we exhibit remarkable behaviours of visual limits of maximal flats in various symmetric spaces of small rank and surprising algebraic restrictions that occur.

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichm\"uller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification of the Torelli space of a hyperbolic surface with marked points, and we show that it is equivariantly isomorphic to the Satake compactification of the image of the period mapping. Finally, we describe the natural stratification of a subset of the boundary.

Let G be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of G, which can be also seen as the space of maximal flats of the symmetric space of G. We define its Chabauty compactification as the closure in the space of closed subgroups of G, endowed with the Chabauty topology. We show that when the real rank of G is 1, or when G=SL3(R) or SL4(R), this compactification is the set of all closed connected abelian subgroups of dimension the real rank of G, with real spectrum. And in the case of SL3(R), we study its topology more closely and we show that it is simply connected.

The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. Let X be a symmetric space of noncompact type, and G be its group of isometries. The space X identifies with the subspace of maximal compact subgroups of G: taking the closure gives rise to the Chabauty compactification of the symmetric space X. Using simpler arguments than those present in the book of Guivarc'h, Ji and Taylor, we describe the subgroups that appear in the boundary of the compactification.

The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.

- "Group actions on injective spaces and Helly graphs", notes for a minicourse at the CRM Montreal, June 2023, arXiv.
- "La conjecture du K(π,1) pour les groupes d'Artin affines (d'après Paolini et Salvetti)", in French, notes for a Bourbaki talk in June 2022, arXiv. Here is a link to the slides of the presentation.

These are lecture notes for a minicourse on group actions on injective spaces and Helly graphs, given at the CRM Montreal in June 2023. We review the basics of injective metric spaces and Helly graphs, emphasizing the parallel between the two theories. We also describe various elementary properties of groups actions on such spaces. We present several constructions of injective metric spaces and Helly graphs with interesting actions of many groups of geometric nature. We also list a few exercises and open questions at the end.

Consider an affine Coxeter group W acting by isometries on the Euclidean space R^n, and the arrangement of its reflection hyperplanes. The fundamental group of the complement Y_W of the complexification of this arrangement in C^n mod out by W is the affine Artin group G_W associated with W. The K(π,1) conjecture states that Y_W is a classifying space for G_W. It has been recently proved by Paolini and Salvetti building on the works of McCammond and Sulway. We will present some ingredients of the proof that rests on the study of dual Garside structures for affine Artin groups, the factorisations of Euclidean isometries, and the shellability of noncrossing partitions. One consequence is that affine Artin groups, as well as braided crystallographic groups, have a finite classifying space.

- "Géométrie des groupes d'Artin, des réseaux et des espaces hyperboliques", Habilitation memoir (December 9th 2021), TEL link, reviewers: Yves Benoist, Martin R. Bridson and Cornelia Drutu.
- "Compactifications géométriques dans les groupes, les espaces symétriques et les immeubles" (theses.fr link) PhD manuscript, advisor: Frédéric Paulin (December 9th, 2011), Referees: Marc Burger and Bertrand Rémy.
- "Compactification de Chabauty des espaces symétriques de type non compact", Master Studies memoir, advisor: Frédéric Paulin (2008).
- "Fonctions spéciales et théorie des représentations", with Michal Was, ENS Paris first year Master studies manuscript, advisor: Charles Torossian (2006).

- Mireille Soergel, at the IMB Dijon, codirection with Luis Paris, defended her PhD in May 2023. Now at the Max Planck Institute in Leipzig.

- Hermès Lajoinie-Dodel, at the University of Montpellier, since October 2021.

- Aitor Azemar
- Indira Chatterji
- François Dahmani
- Vaibhav Gadre
- Sébastien Gouëzel
- Vincent Guirardel
- Nima Hoda
- Camille Horbez
- Jingyin Huang
- Dawid Kielak
- Urs Lang
- Jean Lécureux
- Pablo Lessa
- Joseph Maher
- Jon McCammond
- Masato Mimura
- Damian Osajda
- Harry Petyt
- Catherine Pfaff
- Piotr Przytycki
- Anna-Sofie Schilling
- Petra Schwer
- Caglar Uyanik
- Cormac Walsh
- Anna Wienhard

- Ohio State University, USA, with Jingyin Huang, November 2022.
- UC Santa Barbara, USA, with Jon McCammond, October 2022.
- IRL CRM, Université de Montréal, Canada, CNRS délégation 2022-2023.
- University of Vienna, Austria, with Chris Cashen, March 2019.
- EPFL, Switzerland, with Masato Mimura, May 2018.
- University of Lorraine, Nancy, with Bruno Duchesne, December 2016.
- University of Heidelberg, with Anna Wienhard, May 2015.
- Research in Pairs at the MFO Oberwolfach with Dawid Kielak and Petra Schwer, May 2014.
- University of Heidelberg, with Anna Wienhard, December 2012.
- University of Princeton, with Anna Wienhard, July 2012.