Schedule for: 23w5034 - Interactions between Algebraic Topology and Geometric Group Theory

For a curve complex $C$ of a finite type surface $S$ we construct a tower of hyperbolic $Mod(S)$-complexes $$ C=C_N\to C_{N-1}\to\ldots\to C_1->C_0 $$ such that C_0 is a quasi-tree, all maps are Lipschitz, equivariant, and coarsely onto, and the coarse fibers of each map are quasi-trees. This is a strong version of the fact that curve complexes have finite asymptotic dimension, originally due to Bell-Fujiwara. The result was previously announced by Ursula Hamenstadt. This is joint work with Ken Bromberg and Alex Rasmussen.

Let $Mod(S)$ be the  mapping class group of a connected surface $S$ of finite type with negative Euler characteristic. In joint work with León Álvarez and Sánchez Saldaña, we show that the commensurator of any abelian subgroup of $Mod(S)$ can be realized as the normalizer of a subgroup in the same commensuration class. As a consequence, we give an upper bound for the virtually abelian dimension of $Mod(S)$. These results generalize work by Juan-Pineda–Trujillo-Negrete and Nucinkis–Petrosyan for the virtually cyclic case. In this talk we will introduce the necessary definitions and explain how these results are obtained.

A measure of complexity for topological spaces is given by the minimal number of ``small” open subsets needed to cover the space. For aspherical spaces, we determine this complexity using methods from equivariant topology. As special cases, we recover the (virtual) cohomological dimension, amenable category, and Farber’s topological complexity.

For a topological group $G$ one can topologize the set of homomorphisms $Hom(\mathbb{Z}^n, G)$ as the subspace of $G^n$ consisting of $n$-tuples of pairwise commuting elements of $G$. These spaces of commuting tuples form a simplicial subspace of the usual simplicial space model for the classifying space $BG$. The geometric realization of this simplicial subspace is called the classifying space for commutativity of $G$ and denoted by $Bcom{G}$. The total space of the universal transitionally commutative principal $G$-bundle is denoted $Ecom{G}$. These spaces were introduced by Adem and Cohen. It is fair to say that most of the attention given to $Bcom{G}$ and $Ecom{G}$ so far has been in the case of Lie groups, and more particularly compact Lie groups, but the definitions are also interesting for discrete groups. For discrete groups one can give a simple combinatorial model of the homotopy type of $Ecom{G}$ as a poset, namely, $Ecom{G}$ is homotopy equivalent to the geometric realization of the order complex of the poset of cosets of abelian subgroups of $G$. In this talk we will say how this poset can be studied for fundamental groups of orientable 3-manifold that admit one of the eight 3-dimensional geometries given by Thurston. This is an ongoing work with Omar Antolín-Camarena and Luis Eduardo García-Hernández.

The visual boundary of a hyperbolic space is a quasi-isometry invariant that has proven to be a very useful tool in geometric group theory. In particular, there is a well- defined notion of the visual boundary of a hyperbolic group. When one considers CAT(0) spaces, however, the situation is more complicated, because the visual boundary is not a quasi-isometry invariant. Instead, one can consider a certain subspace of the visual boundary, called the (sublinearly) Morse boundary. In this talk, I will describe a new topology on this boundary and use it to show that the Morse boundary with the restriction of the visual topology is a quasi-isometry invariant in the case of (nice) CAT(0) cube complexes. This result is in contrast to Cashen’s result that the Morse boundary with the visual topology is not a quasi-isometry invariant of CAT(0) spaces in general. This is joint work with Merlin Incerti-Medici.

The surface Houghton group $B_n$ is defined as the asymptotically rigid mapping class group of a surface with exactly n ends, all of them non-planar. The groups $B_n$ are analogous to, and in fact contain, the braided Houghton groups. They fall somewhere between the classical mapping class groups and big mapping class groups. I will outline the proof that these groups have the same finiteness properties as the corresponding Houghton groups.

Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type $F_n$ if it has a classifying space with finitely man cells of dimension at most $n$, generalising finite presentability, which is equivalent to type $F_2$. Hyperbolic groups are of type $F_n$ for all $n$ and it is natural to ask if their subgroups inherit these strong finiteness properties. We use methods from Complex Geometry to show that every uniform arithmetic lattice with positive first Betti number in $PU(n,1)$ contains a subgroup of type $F_{n−1}$ and not $F_n$. This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py.

(joint work with A.Rasmussen and C.Abbott) Recent papers of Balasubramanya and Abbott-Rasmussen have classified the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace-Cornulier-Monod-Tessera. This machinery applies in particular to solvable groups with virtually cyclic abelianizations.

According to the Wikipedia "if proof theory is about the sacred, then model theory is about the profane". In this talk we will present applications of Fraïssé theory to big mapping class groups and to homeomorphism groups of countable spaces.

I will first give a review on the calculations of homology of mapping class groups for finite type surfaces. After that I will discuss what is the story in the case of infinite type surfaces. In particular, I will discuss how one can calculate the homology of mapping class groups for some well-known surfaces, including the disk minus Cantor set. This is based on joint works with Martin Palmer.

We'll explore the problem of finding 'nice' models for the classifying spaces for certain quotients of cubulated groups, and we'll discuss the framework that provides the necessary tools to do so - cubical small-cancellation theory.

The Geometrization Theorem states that every closed 3-manifold can be canonically decomposed into pieces that each have one of eight Thurston geometries. In joint preliminary work with D. Cooper and L. Mavrakis, we show that each of the eight geometries is “locally combinatorially defined” (LCD), which we use to describe a new algorithm for finding a geometric decomposition of a 3-manifold and a new way to build a random 3-manifold. In this talk, I will provide examples to motivate the definition of a family of spaces being LCD. I will then explain how branched 3-manifolds play a key role in our approach towards this problem by sketching the proof of our results for spherical manifolds.

Let $ \Gamma$ be a hyperbolic group and assume there is a manifold model $M$ for its classifying space. The Nielsen realization problem asks when a group $G < Out(\Gamma)$ can be realized by a $G$-action on $M$. We will survey some old and new results on this problem.

I will describe the  $C^{0}$- isometric version of  the  Zimmer  program  for three dimensional manifolds and  closely  related three dimensional   geodesic metric  spaces  like  geometric Orbifolds  and Alexandrov  spaces. Joint  with Manuel Sedano-Mendoza.