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(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 ...

20F65 ; 57M07

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Using a very recent result of Calderon and Salter, we relate small type Artin groups defined by Coxeter diagram which are trees to mapping class groups. This gives information on both the Artin groups with respect to commensurability and hyperbolicity of the parabolic subgroup graph as well as information on the mapping class group and its associated geometric spaces, namely generating sets of finite index subgroups and fundamental groups of strata of abelian differentials. I’ll try to highlight the many ways in which this reflects various aspects of Mladen’s work.
Using a very recent result of Calderon and Salter, we relate small type Artin groups defined by Coxeter diagram which are trees to mapping class groups. This gives information on both the Artin groups with respect to commensurability and hyperbolicity of the parabolic subgroup graph as well as information on the mapping class group and its associated geometric spaces, namely generating sets of finite index subgroups and fundamental groups of ...

20F65 ; 53C24 ; 57S25

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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 ...

20F65 ; 57M07

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Multi angle  Counting curves of given type, revisited
Souto, Juan (Auteur de la Conférence) | CIRM (Editeur )

Mirzakhani wrote two papers studying the asymptotic behaviour of the number of curves of a given type (simple or not) and with length at most $L$. In this talk I will explain a new independent proof of Mirzakhani’s results.
This is joint work with Viveka Erlandsson.

57N05 ; 30F45 ; 30F60 ; 32G15 ; 57M50

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We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional CAT(0) complex has a fixed point, then the action is trivial. In particular, all actions of finitely generated torsion groups on such complexes are trivial. As an ingredient, we prove that the image of an immersed loop in a graph of girth 2π with length not commensurable to π has diameter > π. This is related to a theorem of Dehn on tiling rectangles by squares.
This is joint work with Sergey Norin and Damian Osajda.
We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional CAT(0) complex has a fixed point, then the action is trivial. In particular, all actions of finitely generated torsion groups on such complexes are trivial. As an ingredient, we prove that the image of an immersed loop in a graph of girth 2π with length not commensurable to π has diameter > π. This is related to a theorem of Dehn on ...

20F65

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