Tag - Geometric group theory

Emily Stark: Action rigidity for free products of hyperbolic manifold groups

7th May, 2020

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold is not action rigid for all n ≥ 3. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Paul Schupp: Coarse computabilty and the Hausdorff distance between Turing degrees.

9th April, 2020

Coarse computabilty studies how well arbitrary sets can be approximated in terms of computable sets. Define two sets A and B of natural numbers to be coarsely similar, written A∼cB, if their symmetric difference A Δ B has density 0 in the sense of classical asymptotic density from number theory. This relation is an equivalence relation, so we consider the space 𝒮 = 𝒫(ℕ)/∼c of coarse similarity classes. There is a natural density metric defined on 𝒮 by setting δ(A,C) to be the upper density of their symmetric difference.

The space 𝒮 is very interesting. While neither separable nor compact, it is both complete and contractible. Indeed, 𝒮 is a geodesic metric space so it is a hyperbolic space in the sense of Gromov with the property that there are uncountably many different geodesics between any two distinct points of 𝒮. Define the core, or lower cone, κ(d), of a Turing degree d to be the family { [A] } of all classes of sets such that A ≤T d. The closure d̅ of the degree d is the closure of κ(d) in 𝒮.

Define the distance H(d,e) between two Turing degrees as the Hausdorff distance between their closures in 𝒮. This distance has an equivalent definition solely in terms of computability theory. It turns out that the the Hausdorff distance between any two degrees is either 0,1/2 or 1. This is joint work with Carl Jockusch.

Alina Vdovina: Buildings, C*-algebras and new higher-dimensional analogues of the Thompson groups

2nd April, 2020

We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory.

The underlying building structure allows explicit computation of the K-theory. We will also present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomophic groups.

Doron Puder: Word-Measures on Unitary Groups

2nd February, 2017

One approach to studying properties of random walks on groups with random generators is to study word-measures on these groups. This approach was proven useful for the study of symmetric groups and random regular graphs. In the current work we focus on the unitary groups U(n). For example, if w is a word in F2 = <x,y>, sample at random two elements from U(n), A for x and B for y, and evaluate w(A,B). The measure of this random element is called the w measure on U(n). We study the expected trace (and other invariants) of a random unitary matrix sampled from U(n) according to the w-measure, and find surprising algebraic properties of w that determine these quantities.

Federico Rodriguez Hertz: Rigidity of hyperbolic higher rank lattice actions

3rd December, 2013

I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, SL(n,ℤ) acting on 𝕋n, and try to present the ideas of the proof. The result imply existence of invariant measures for SL(n,ℤ) actions on 𝕋n with standard homotopy data as well as global rigidity of Anosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure.

Hee Oh: Counting for orbits of thin groups, via ergodic methods

7th February, 2012

For a geometrically finite hyperbolic group with small critical exponent, the spectral
method for counting is not available, as there is no point eigenvalue of the Laplace operator on the L2-spectrum. We will explain counting results for orbits of a big class of thin groups acting on a symmetric variety of the real hyperbolic group, which are obtained via ergodic approach.

Hee Oh: Effective circle count for Apollonian circle packings, via spectral methods

6th February, 2012

We will describe a recent effective counting result for Apollonian circle packings. The main ingredient of this result is an effective equidistribution of closed horospheres in an infinite volume hyperbolic 3-manifold whose fundamental group has critical exponent bigger than one. We will explain how the spectral theory of Lax and Phillips can be used for such equidistribution results.