Tag - Group theory

Frank Wagner: Torsion Subgroups of Groups with Cubic Dehn Function

23rd April, 2020

The Dehn function of a finitely presented group, first introduced by Gromov, is a useful invariant that is closely related to the solvability of the group’s word problem. It is well-known that a finitely presented group is word hyperbolic if and only if it has sub-quadratic (and thus linear) Dehn function. A result of Ghys and de la Harpe states that no word hyperbolic group can have a (finitely generated) infinite torsion subgroup. We show that the same does not hold for finitely presented groups with Dehn function as small as cubic. In particular, for every m≥2 and sufficiently large odd integer n, there exists an embedding of the free Burnside group B(m,n) into a finitely presented group with cubic Dehn function.

Sean Eberhard: An asymptotic for the Hall-Paige conjecture

20th April, 2020

A complete mapping of a group G is a bijection f : G → G such that the map x f(x) is also a bijection. Hall and Paige conjectured in 1955 that every finite group satisfying a certain necessary condition has a complete mapping; this was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups. I will discuss recent joint work with Freddie Manners and Rudi Mrazovic in which we asymptotically count complete mappings using something like the circle method.

Tatiana Smirnova-Nagnibeda: Spectra of Laplacians on Cayley and Schreier graphs

16th April, 2020

We are interested in Laplacians on graphs associated with finitely generated groups: Cayley graphs and, more generally, Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general. We will discuss various techniques that allow to construct examples with different types of spectra - connected, union of two intervals, totally disconnected - and with various types of spectral measure. The problem of spectral rigidity will also be addressed.

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.

Zoran Šunić: Deciding if a right-angled Artin group is free-by-free is NP-complete

19th March, 2020

We show that deciding if a right-angled Artin group is free-by-free is an NP-complete problem. The work is based on an earlier result by Susan Hermiller and the speaker stating that the right-angled Artin group AΓ defined by the graph Γ is free-by-free if and only if Γ is 2-breakable (a graph Γ is 2-breakable if there exists an independent set D of vertices in Γ such that every cycle in Γ contains as least two vertices from D). We reduce the 3SAT Problem to the problem of deciding if a given graph is 2-breakable (in fact, k-breakable, for any fixed k ≥ 1). Once it is shown that the problem is NP-complete, it is not difficult to show that it stays NP-complete even if we restrict it to right-angled Artin groups defined by planar graphs. Note that the more special problem of deciding if a right-angled Artin group is free-by-infinite-cyclic has a very simple answer. Namely, it follows easily from known results that the following three statements are equivalent. (1) AΓ is free-by-infinite-cyclic. (2) Γ is a forest. (3) AΓ embeds in the right angled group defined by the path of length 3. (Joint work with David Carroll and Benjamin Francisco.)

Junho Peter Whang: Non-linear descent on moduli of local systems

31st October, 2017

In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of non-linear descent. Generalizing this, we consider families of log Calabi-Yau varieties arising as moduli spaces for local systems on topological surfaces, and prove a structure theorem for their integral points using mapping class group dynamics. The result is reminiscent of the finiteness of class numbers for linear arithmetic group actions on homogeneous varieties, and this Diophantine perspective guides us to obtain new extensions of classical results on hyperbolic surfaces along the way.