Geometry and Topology Seminar

Monday August 17, 15:00, room MS.03

Joseph Maher (CUNY)

Random walks on groups with hyperbolic properties

Abstract: (Joint work with Giulio Tiozzo.) We shall introduce Gromov hyperbolic spaces and groups, and give some examples. We shall also discuss more general examples of groups which are not hyperbolic but act by isometries on (not necessarily proper) Gromov hyperbolic spaces, which include acylindrical groups such as the mapping class group and $\Out(F_n)$. We shall discuss convergence to the boundary for random walks on hyperbolic groups, and then indicate how this may be extended to the more general case. Applications include linear progress, sublinear tracking, and identifying the Poisson boundary for acylindrical groups.

Thursday October 15, 15:00, room MS.04

Federica Fanoni (Warwick)

Systems of curves and sytoles on surfaces

Abstract: Given a surface, it is natural to study systems of curves with some topological condition, such as bounds on the number of intersections. If the surface is endowed with a hyperbolic structure, we can ask what changes if we also require all curves to be systoles (shortest closed geodesics). In this talk I will discuss two problems in this context: bounding the size of k-systems and of k-filling sets. This is joint work with Hugo Parlier.

Thursday October 22, 15:00, room MS.04

Francesca Iezzi (Warwick)

Projections of spheres to arc graphs

Abstract: Sphere graphs were introduced by Hatcher in the 1990's as a kind of analogue of curve graphs for 3-manifolds. These objects have turned out to be a useful tool in the study of outer automorphism groups of free groups. Under certain hypothesis, there is a natural injective map $\phi$ of the arc graph of a surface with boundary into the sphere graph of a 3-manifold. In the talk, after giving the definition of sphere graphs and arc graphs, and explaining the definition of $\phi$, I will prove that $\phi$ admits a coarsely defined Lipschitz left inverse.

Thursday October 29, 15:00, room MS.04

Martin Bridson (Oxford)

The symmetries of the free factor complex

Abstract: I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a free group $F$ is exactly $\Aut(F)$, provided that $F$ has rank at least three. The free factor complex was introduced by Hatcher and Vogtmann as the natural analogue in the free setting of the classical building that is the complex of summands for $\ZZ^n$: the inclusion of $\ZZ^n$ into $\QQ^n$ induces an isomorphism from the poset of summands to the spherical building for $\GL(n,\QQ)$. If $n > 2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of this building is $\PGL(n,\QQ)$, which has index two in the abstract commensurator of $\GL(n,\ZZ) = \Out(\ZZ^n)$. In the free setting, $\Out(F)$ is its own commensurator (I shall discuss this) and our theorem appears as the appropriate analogue of the FTPG.

Thursday November 5, 15:00, room MS.04

Russel Lodge (Jacobs)

Boundary values of Thurston's pullback map

Abstract: In his fundamental theorem of holomorphic dynamics, W. Thurston characterizes those postcritically finite topological branched covers from the two-sphere to itself that are equivalent to rational maps. This so-called "Thurston equivalence" was not very well understood until Bartholdi and Nekrashevych, in solving the "Twisted rabbit problem", produced a powerful algebraic invariant using the theory of iterated monodromy groups. I will show that the group theory has further implications for the iteration on Teichmüller space used by Thurston to prove his theorem, as well as the dynamics of multicurves under iterative preimage by the branched cover.

Thursday November 19, 15:00, room MS.04

Bram Petri (Max-Planck)

Random surfaces

Abstract: Random surfaces can be used to study the geometric properties of typical (hyperbolic) surfaces of large genus. In this talk, a random surface will be a surface obtained by randomly gluing together triangles along their sides. I will explain how these surface relate to random regular graphs and how that can be used to study the lengths of curves on random surfaces.

Thursday November 26, 15:00, room MS.04

Viveka Erlandsson (Aalto, Fribourg)

Counting curves on hyperbolic surfaces

Abstract: Let $c$ be a closed curve on a hyperbolic surface $S = S(g,n)$ and let $N_c(L)$ denote the number of curves in the mapping class orbit of $c$ with length bounded by $L$. Due to Mirzikhani it is know that in the case that $c$ is simple this number is asymptotic to $L^{6g-6+2n}$. In this talk I will discuss the case when $c$ is an arbitrary closed curve, i.e. not necessarily simple. I will show that in the case when $S$ is the punctured torus, the limit as $L$ goes to infinity of $N_c(L)/L^2$ exists, and discuss some results in the case of a general surface. This is joint work with Juan Souto.

Thursday November 26, 16:00, room MS.04

Matthew Durham (Michigan)

A new boundary for the mapping class group

Abstract: I will introduce a geometrically intrinsic compactification of the mapping class group which, in the course of proving theorems, functions like the Gromov boundary of a hyperbolic group. I will discuss some initial applications, including a Rank-Rigidity type theorem and the fact that the boundary is a model for the Poisson boundary for random walks on the mapping class group. Time permitting, I will indicate some future directions. This is joint work with Mark Hagen and Alessandro Sisto.

Thursday December 3, 15:00, room MS.04

Irene Pasquinelli (Durham)

Deligne-Mostow lattices and cone metrics on the sphere

Abstract: Finding lattices in $\PU(n,1)$ has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

Thursday December 10, 15:00, room MS.04

Kai-Uwe Bux (Bielefeld)

Connectivity of arc matching complexes

Abstract: Let $G$ be a finite graph. A matching in $G$ is a set of pairwise disjoint edges. The associated matching complex is the simplicial complex whose vertices are the edges of $G$ and whose simplices are precisely the matchings in $G$. Questions of connectivity and higher connectivity of matching complexes are a staple in topological combinatorics.

Now let $S$ be a connected surface, possibly with boundary, and mark points $p_1$, ..., $p_n$ in $S$ in 1-1 correspondence to the vertices of $G$. An arc matching complex consists of lifts of matchings, i.e. a collection of pairwise disjoint arcs in $S$ between the $p_i$ is a simplex if all these arcs descend to edges in $G$. I will discuss connectivity and higher connectivity of some arc matching complexes that occur as "descending links" in the context of finitiness properties of groups.

Thursday December 10, 16:00, room MS.04

Arnaud Hilion (Aix-Marseille)

Invariant measures for train track towers

Abstract: Train track towers have been used to study the ergodic properties of free group actions on R-trees and on laminations. After discussing the necessary background, I will explain how towers can also be used to investigate the invariant currents of a large class of free group endomorphisms.