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Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. In this talk I will present the proof (joint with Hermiller, Holt and Rees) that the conjugacy growth series of a virtually cyclic group is rational, and then also confirm the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental (joint with Antolín). The result for non-elementary hyperbolic groups can be used to prove a formal language version of Rivin's conjecture for any finitely generated acylindrically hyperbolic group G, namely that no set of minimal length conjugacy representatives of G can be regular.[-]

Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. In this talk I will present the proof (joint with Hermiller, Holt and Rees) that the conjugacy growth series of a virtually cyclic group is rational, and then also confirm the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental (joint ...[+]

20F67 ; 68Q45

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Let $G$ be a torsion-free hyperbolic group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. We want to understand the possible growth types for the word length of $f^n(g)$, where $g$ is an element of $G$. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We address the general case of a torsion-free hyperbolic group $G$; we show that every element in $G$ has a well-defined exponential growth rate under iteration of $f$, and that only finitely many exponential growth rates arise as $g$ varies in $G$. In addition, we show the following dichotomy: every element of $G$ grows either exponentially fast or polynomially fast under iteration of $f$.
This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.[-]

Let $G$ be a torsion-free hyperbolic group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. We want to understand the possible growth types for the word length of $f^n(g)$, where $g$ is an element of $G$. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We address the ...[+]

57M07 ; 20E06 ; 20F34 ; 20F65 ; 20E36 ; 20F67

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Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. If the automorphism is atoroidal, then the resulting free-by-cyclic group is hyperbolic by work of Brinkmann. In addition, if the automorphism is fully irreducible, then work of Kapovich-Kleiner proves the boundary of the group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further study the boundary and large-scale geometry of these groups. In this talk, I will explain how to construct explicit embeddings of non-planar graphs into the boundary of these groups whenever the group is hyperbolic. Along the way, I will illustrate how our methods distinguish free-by-cyclic groups which are the fundamental group of a 3-manifold. This is joint work with Yael Algom-Kfir and Arnaud Hilion.[-]

Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. If the automorphism is atoroidal, then the resulting free-by-cyclic group is hyperbolic by work of Brinkmann. In addition, if the automorphism is fully irreducible, then work of Kapovich-Kleiner proves the boundary of the group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further ...[+]

20F65 ; 20F67 ; 20E36

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The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin. However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions remain unanswered.
In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic structures and thus answer some of the open questions. I will end my talk by discussing ongoing work in the area.
This talk contains some joint work with C.Abbott, D.Osin and A.Rasmussen.[-]

The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin. However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions remain unanswered.
In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic structures and thus answer some of the open questions. I will end my talk by ...[+]

20F65 ; 20F67 ; 20E08

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The theory of group actions on CAT(0) cube complexes has exerted a strong influence on geometric group theory and low-dimensional topology in the last two decades. Indeed, knowing that a group G acts properly and cocompactly on a CAT(0) cube complex reveals a lot of its algebraic structure. However, in general, "cubulations'' are non-canonical and the group G can act on cube complexes in many different ways. It is thus natural to try and formulate a good notion of "space of all cubulations of G'', which would prove useful in the study of Out(G) for quite general groups G. I will describe some results in this direction, based on joint works with J. Beyrer and M. Hagen.[-]

The theory of group actions on CAT(0) cube complexes has exerted a strong influence on geometric group theory and low-dimensional topology in the last two decades. Indeed, knowing that a group G acts properly and cocompactly on a CAT(0) cube complex reveals a lot of its algebraic structure. However, in general, "cubulations'' are non-canonical and the group G can act on cube complexes in many different ways. It is thus natural to try and ...[+]

20F65 ; 20F67 ; 20F34 ; 57-XX

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Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), cocompactly cubulated groups, hyperbolic groups, Coxeter groups and the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. I will discuss some of these examples as well as some general constructions and properties of shortcut graphs and groups.[-]

Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), cocompactly cubulated ...[+]

20F65 ; 20F67 ; 05C12

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The Cremona group is the group of birational transformations of the projective plane. Even if this group comes from algebraic geometry, tools from geometric group theory have been powerful to study it. In this talk, based on a joint work with Christian Urech, we will build a natural action of the Cremona group on a CAT(0) cube complex. We will then explain how we can obtain new and old group theoretical and dynamical results on the Cremona group.

14E07 ; 20F65 ; 20F67

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The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.[-]

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry ...[+]

20F65 ; 20F67 ; 20E06 ; 57M07 ; 57M10

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There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). We will discuss several natural decompositions that arise in the study of rational maps, such as Pilgrim's canonical decomposition and Levy decomposition (by Bartholdi and Dudko). I will also introduce a new decomposition of rational maps based on the topology of their Julia sets (obtained jointly with Dima Dudko and Dierk Schleicher). At the end of the talk, we will briefly consider connections of this novel decomposition to geometric group theory and self-similar groups.[-]

There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). We will discuss several natural decompositions that arise in the study of rational maps, such as Pilgrim's canonical decomposition and Levy decomposition (by Bartholdi and Dudko). I will also introduce a new decomposition of rational maps based on the ...[+]

37F10 ; 37F20 ; 37B10 ; 37B40 ; 20F67 ; 20E08

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I will present an overview and some specific problems concerning appearances of weakly modular graphs in group theory, and particularly, in geometric group theory. It will be based on works joint with Victor Chepoi and other colleagues from metric graph theory.

05C12 ; 05C75 ; 05E45 ; 20F67 ; 51K05

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Documents 20F67 16 résultats

Formal conjugacy growth and hyperbolicity - Ciobanu, Laura (Auteur de la Conférence) | CIRM H Nouveau

Automorphisms of hyperbolic groups and growth - Horbez, Camille (Auteur de la Conférence) | CIRM H Nouveau

The visual boundary of hyperbolic free-by-cyclic groups - Stark, Emily (Auteur de la Conférence) | CIRM H Nouveau

Quasi-parabolic structures on groups - Balasubramanya, Sahana (Auteur de la Conférence) | CIRM H Nouveau

Spaces of cubulations - Fioravanti, Elia (Auteur de la Conférence) | CIRM H Nouveau

Shortcut graphs and groups - Hoda, Nima (Auteur de la Conférence) | CIRM H Nouveau

Action of the Cremona group on a CAT(0) cube complex - Lonjou, Anne (Auteur de la Conférence) | CIRM H Nouveau

Action rigidity for free products of hyperbolic manifold groups - Stark, Emily (Auteur de la Conférence) | CIRM H Nouveau

Decomposition results in rational dynamics - Hlushchanka, Mikhail (Auteur de la Conférence) | CIRM H Nouveau

Weakly modular graphs in group theory - Osajda, Damian (Auteur de la Conférence) | CIRM H Nouveau