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# Documents  57M07 | enregistrements trouvés : 7

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## Post-edited  Automorphisms of hyperbolic groups and growth Horbez, Camille (Auteur de la Conférence) | CIRM (Editeur )

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 ...

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## Post-edited  The conjugacy problem for polynomially growing elements of $Out(F_{n})$ Feighn, Mark (Auteur de la Conférence) | CIRM (Editeur )

(joint work with Michael Handel) $Out(F_{n}) := Aut(F_{n})/Inn(F_{n})$ denotes the outer automorphism group of the rank n free group $F_{n}$. An element $f$ of $Out(F_{n})$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_{n}), n > 2$, of elements with non-linear polynomial growth is a feature of $Out(F_{n})$ not shared by mapping class groups of surfaces.
To avoid some finite order behavior, we restrict attention to the subset $UPG(F_{n})$ of $Out(F_{n})$ consisting of polynomially growing elements whose action on $H_{1}(F_{n}, Z)$ is unipotent. In particular, if $f$ is polynomially growing and acts trivially on $H_{1}(F_{n}, Z_{3})$ then $f$ is in $UPG(F_{n})$ and further every polynomially growing element of $Out(F_{n})$ has a power that is in $UPG(F_{n})$. The goal of the talk is to describe an algorithm to decide given $f,g$ in $UPG(F_{n})$ whether or not there is h in $Out(F_{n})$ such that $hf h^{-1} = g$.
The conjugacy problem for linearly growing elements of $UPG(F_{n})$ was solved by Cohen-Lustig. Krstic-Lustig-Vogtmann solved the case of linearly growing elements of $Out(F_{n})$.
A key technique is our use of train track representatives for elements of $Out(F_{n})$, a method pioneered by Bestvina-Handel in the early 1990s that has since been ubiquitous in the study of $Out(F_{n})$.
(joint work with Michael Handel) $Out(F_{n}) := Aut(F_{n})/Inn(F_{n})$ denotes the outer automorphism group of the rank n free group $F_{n}$. An element $f$ of $Out(F_{n})$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_{n}), n > 2$, of elements with non-linear polynomial growth is a feature of $Out(F_{n})$ not shared by mapping class groups of ...

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## Multi angle  Quantum symmetry of conformal blocks andrepresentations of braid groups at roots of unity Kohno, Toshitake (Auteur de la Conférence) | CIRM (Editeur )

braid groups - conformal blocks - KZ equation - quantum group symmetry - hypergeometric integrals - Gauss-Manin connection

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## Multi angle  Groups with Bowditch boundary a 2-sphere Tshishiku, Bena (Auteur de la Conférence) | CIRM (Editeur )

Bestvina-Mess showed that the duality properties of a group $G$ are encoded in any boundary that gives a Z-compactification of $G$. For example, a hyperbolic group with Gromov boundary an $n$-sphere is a PD$(n+1)$ group. For relatively hyperbolic pairs $(G,P)$, the natural boundary - the Bowditch boundary - does not give a Z-compactification of G. Nevertheless we show that if the Bowditch boundary of $(G,P)$ is a 2-sphere, then $(G,P)$ is a PD(3) pair.
This is joint work with Genevieve Walsh.
Bestvina-Mess showed that the duality properties of a group $G$ are encoded in any boundary that gives a Z-compactification of $G$. For example, a hyperbolic group with Gromov boundary an $n$-sphere is a PD$(n+1)$ group. For relatively hyperbolic pairs $(G,P)$, the natural boundary - the Bowditch boundary - does not give a Z-compactification of G. Nevertheless we show that if the Bowditch boundary of $(G,P)$ is a 2-sphere, then $(G,P)$ is a ...

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## Multi angle  The Farrell-Jones conjecture for free-by-cyclic groups Bestvina, Mladen (Auteur de la Conférence) | CIRM (Editeur )

The Farrell-Jones conjecture for a given group is an important conjecture in manifold theory. I will review some of its consequences and will discuss a class of groups for which it is known, for example 3-manifold groups. Finally, I will discuss a proof that free-by-cyclic groups satisfy FJC, answering a question of Lück.
This is joint work with Koji Fujiwara and Derrick Wigglesworth.

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## Multi angle  Homomorphisms to 3-manifold groups and other families Groves, Daniel (Auteur de la Conférence) | CIRM (Editeur )

We are interested in the structure of the set of homomorphisms from a fixed (but arbitrary) finitely generated group G to the groups in some fixed family (such as the family of 3-manifold groups). I will explain what one might hope to say in different situations, and explain some applications to relatively hyperbolic groups and acylindrically hyperbolic groups, and some hoped-for applications to 3-manifold groups.
This is joint work with Michael Hull and joint work in preparation with Michael Hull and Hao Liang.
We are interested in the structure of the set of homomorphisms from a fixed (but arbitrary) finitely generated group G to the groups in some fixed family (such as the family of 3-manifold groups). I will explain what one might hope to say in different situations, and explain some applications to relatively hyperbolic groups and acylindrically hyperbolic groups, and some hoped-for applications to 3-manifold groups.
This is joint work with Michael ...

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## Multi angle  Conformal dimension and free by cyclic groups Algom-Kfir, Yael (Auteur de la Conférence) | CIRM (Editeur )

Let $G$ be a hyperbolic group. Its boundary is a topological invariant within the quasi-isometry class of $G$ but it is far from being a complete invariant, e.g. a random group at density ¡1/2 is hyperbolic (Gromov) and its boundary is homeomorphic to the Menger curve (Dahmani-Guirardel-Przytycki) but Mackay proved that there are infinitely many quasi-isometry classes of random groups at density d for small enough d.
We discuss the conformal dimension of a hyperbolic group, a quasi-isometry invariant introduced by Pansu. Paulin proved that this is a complete $QI$ invariant of the group. We discuss a technique of Pansu and Bourdon for bounding the conformal dimension from below. We then relate this technique to the family of hyperbolic free by cyclic groups. This is work in progress towards the ultimate goal of showing that there are infinitely many $QI$ classes of free by cyclic groups.
This is joint work with Bestvina, Hilion and Stark
Let $G$ be a hyperbolic group. Its boundary is a topological invariant within the quasi-isometry class of $G$ but it is far from being a complete invariant, e.g. a random group at density ¡1/2 is hyperbolic (Gromov) and its boundary is homeomorphic to the Menger curve (Dahmani-Guirardel-Przytycki) but Mackay proved that there are infinitely many quasi-isometry classes of random groups at density d for small enough d.
We discuss the conformal ...

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