# Geometric Topology Fair 2017

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# Abstracts

Boundary amenability for Out(F_n)

Abstract: The motivation for the talk is the recent result, joint with Vincent Guirardel and Camille Horbez, that Out(F_n) admits a topologically amenable action on a Cantor set. This implies the Novikov conjecture for Out(F_n) and its subgroups. Most of the talk will be an introduction to boundary amenability and ways to prove it for simpler groups.

Embeddability between RAAGs
Sang Jin Lee (Konkuk University)

Abstract: For a finite simplicial graph $\Gamma$, let $G(\Gamma)$ denote the right-angled Artin group on the complement graph of $\Gamma$. (Non-adjacent vertices of $\Gamma$ correspond to commutative generators in $G(\Gamma)$.) In the talk, we will discuss the following problem: for which graph $\Lambda$ does there exist an embedding of $G(\Gamma)$ into $G(\Lambda)$? Main interest is when $\Lambda$ is a tree or a path graph. This is joint work with Eon-Kyung Lee.

Diophantine approximation and homogeneous dynamics
Seonhee Lim (Seoul National University)

Abstract: In this talk, we will review a few known results on Diophantine approximation which are obtained by homogeneous dynamics. Then we will focus on one problem related to inhomogeneous Diophantine approximation : Hausdorff dimension of bad grids. (This is a joint work with U. Shapira and N. de Saxce.)

Fibered commensurability on Out(F_n)
Hidetoshi Masai (Tohoku University)

Abstract:Calegari-Sun-Wang defined a covering relation on the set of mapping classes of surface automorphisms. A mapping class $\phi$ covers $\psi$ if $\phi$ is a lift of a power of $\psi$ with respect to a finite covering of underlying surfaces. Commensurability given by this covering is called fibered commensurability. In this talk, we analogously define a covering relation on the set of outer automorphisms of free groups, and discuss its properties. This is a joint work with Ryosuke Mineyama.

Discontinuous motions of Cannon-Thurston maps
Kenichi Ohshika (Osaka University)

Abstract: In one of the famous 24 problems of Thurston which appeared in Bull. AMS in 1982, he asked if Cannon-Thurston maps move continuously in the deformation spaces of Kleinian surface groups. Mj and Series gave an example of convergent sequence of quasi-Fuchsian groups whose Cannon-Thurton maps do not converge pointwise to that of the limit group. In this talk, we shall give a complete criterion for C-T maps of quasi-Fuchsian groups to converge pointwise. Furthermore, we shall determine at which points the convergence breaks down if the criterion does not hold. This is joint work with Mahan Mj of Tata Institute.

Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces
Takuya Sakasai (University of Tokyo)

Abstract: We construct an abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra generated by the fundamental representation of the symplectic group. It gives an alternative proof of the fact first shown by Bartholdi that the top rational homology group of the moduli space of metric graphs of rank 7 is one dimensional. As an application, we construct a non-trivial abelian quotient of the homology cobordism group of a surface of positive genus. This talk is based on joint works with Shigeyuki Morita, Masaaki Suzuki and Gwénaël Massuyeau.

Fast computation in mapping class groups
Balazs Strenner (Georgia tech)

Abstract: The talk will be on a project, joint with Dan Margalit and Oyku Yurttas, whose goal is to give a framework for fast computation in mapping class groups. We show that there is a quadratic-time algorithm that computes the Nielsen-Thurston type of a mapping class (finite order, pseudo-Anosov or reducible). It also finds the reducing curves and the stretch factors and invariant foliations on pseudo-Anosov components.

Constructions of hyperbolic surfaces with long systoles.
Bram Petri (University of Bonn)

Abstract: The length of the shortest non-contractible curve - the systole - of a closed hyperbolic surface is at most logarithmic in the genus of the surface. In this talk I will speak about constructions of hyperbolic surfaces with systoles of logarithmic length and discuss their properties. Part of this is joint work with Alex Walker.