# Fall2018

**All the talks are 4:00pm-5:00pm in Room 5417 at the CUNY Graduate Center.**

**Wine and cheese are served afterwards in the math lounge on the 4th floor.**

Sep. 7

### Tullio Ceccherini-Silberstein (Universita degli Study del Sannio)

TITLE: Garden of Eden Theorems: from Symbolic Dynamics to Algebraic

Dynamical Systems.

ABSTRACT: The Garden of Eden Theorem is a central result in the theory of

cellular automata. Given a finite alphabet set A and a group G, a

continuous G-equivariant (w.r. to the G-shift) map $\tau \colon A^G \to

A^G$ is called a cellular automaton. The GOE theorem states that a

cellular automaton is surjective if and only if it is pre-injective (a

weaker condition than injectivity).

It was proved by Moore and Myhill in 1963 with G = Z^d the free abelian

group of rank d, and it was extended to all amenable groups

(Ceccherini-Silberstein, Machi, and Scarabotti 1999; Gromov 1999).

It was later shown that it fails to hold for any nonamaneble group

(Bartholdi, 2010), thus yielding a new characterization of amenability.

Following a suggestiion by Gromov, namely that the Garden of Eden theorem

could be extended to dynamical systems with a suitable hyperbolic flavor,

a Garden of Eden type theorem was proved for Anosov diffeomorphisms on

tori (Ceccherini-Silberstein and Coornaert, 2015) and for principal algebraic

dynamical systems satisfying a weak form of expansivity

(Ceccherini-Silberstein, Coornaert, and Li, 2018).

The talk will be completely self-contained.

Sep. 14 Martin Kreuzer

"Computing the Canonical Decomposition of a Finite Z-Algebra".

Abstract: In this joint work with Alexei Miasnikov and Florian Walsh,

we present an efficient method for computing the canonical decomposition

of a finite Z-algebra R into irreducible factors. Although the existence

of this decomposition has been known for along time, its equational

definability

is a more recent result. The first step is to compute the maximal ring

of scalars S(R). This is a finite commutative Z-algebra with an

explicitly computable

presentation. Then we use strong Groebner bases to calculate the fundamental

idempotents of S(R),. They allow us to compute the desired canoncial

decomposition in the final step. All steps of the procedure have a

polynomial

complexity, except for the strong Groebner basis calculation which

has a singly exponential bound.

Sep.28 Vladimir Shpilrain, Complexity in SL_2(Z) and SL_2(Q)

Abstract. We reflect on how to define complexity of a matrix and how to sample a random invertible matrix. We also discuss a related issue of complexity of algorithms in matrix groups, focusing on computational complexity of the subgroup membership problem for some important special subgroups of SL_2(Z) and SL_2(Q).

The talk is based on joint work with Anastasiia Chorna, Katherine Geller, Lisa Bromberg, and Alina Vdovina.

Oct.5 Alexei Miasnikov

Malcev's Problems and First-Order Paradise in Groups

Oct.12 Ilya Kapovich (Hunter College, CUNY)

Title: Counting conjugacy classes of fully irreducibles: double exponential growth

Abstract: A 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface S of genus $g\ge 2$, the number $N(L)$ of closed Teichmuller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi\in MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)$ is the ``dilatation" or ``stretch factor" of $\phi$. We consider an analogous problem in the $Out(F_r)$ setting for the number $N_r(L)$ of $Out(F_r)$conjugacy classes of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. We prove, for $r\ge 3$, that $N_r(L)$ grows doubly exponentially in $L$ as $L\to\infty$, in terms of both lower and upper bounds. These bounds reveal behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff.

Oct.19 Daniel Studenmund (U of Notre-Dame)

Title: Commensurability growth of nilpotent groups

Abstract: A classical area of study in geometric group theory is subgroup growth, which counts the number of subgroups of a given group Gamma as a function their index. We will study a richer function, the commensurability growth, associated to a subgroup Gamma in an ambient group G. The main results of this talk concern arithmetic subgroups Gamma of unipotent groups G, following subgroup growth results by Grunewald, Segal, and Smith. We start with the simplest example of the integers in the real line. This is joint work with Khalid Bou-Rabee.

Oct.26 Lam Pham (Yale University)

**Title: On Uniform Kazhdan Constants for Finitely Generated Linear Groups. **

**Abstract:** If

is a finitely generated group and is a unitary representation of on a Hilbert space without -invariant vectors, it is of interest to know if has a spectral gap; when all such representations

have a spectral gap, is said to have Kazhdan's Property . in general these spectral gaps depend on the choice of the generating set of

, and an important question is whether this dependence on can be removed. It is an open problem to determine if is uniform Kazhdan (i.e., the Kazhdan constant is independent of the choice of generators of

). In this talk, I will: (1) give an overview of the literature on explicit Kazhdan constants of finitely generated groups since the first explicit computation due to Burger (1991), and (2) present some new results on uniform spectral gaps for actions of the affine group over the integers.

Nov.2 Ben Fine's 70 conference

Nov.9 Catherine Pfaff (Queen's University at Kingston**)**

**Title: Random automorphisms of free groups and what happens when you iterate them.**

Abstract:

Two of the most natural and interesting questions one can ask about an automorphism group is what a random element of the automorphism group looks like and what happens as one repeatedly applies the automorphism to an element of the group (the asymptotic conjugacy class invariants). In the mapping class group circumstance, these questions (and their intersection) have been thoroughly studied with results dating back to Nielsen and Thurston, and then more recently with Dahmani, Horbez, Maher, Rivin, Sisto, Tiozzo, etc. While some is known in the outer automorphism group of the free group setting, little to nothing has been known about the most basic questions in the intersection of the main classes of questions, i.e. understanding the asymptotic conjugacy class invariants of random (outer) automorphisms of free groups. Together with Ilya Kapovich, Joseph Maher, and Samuel Taylor, we give a fairly detailed answer to this question.

Nov.16 Ben Steinberg (CCNY) Title: Homological Finiteness Conditions for One-relator Monoids

Abstract: We announce in this talk a positive solution to Kobayashi's 2000 question whether every one-relator monoid is of type $FP_\infty$. We sketch the main ideas of the proof, which are topological in nature. This is joint work with Robert Gray.

Nov.23 Thanksgiving

Nov.30 Khalid Bou-Rabee (CCNY) Title: On local residual finiteness of abstract commensurators of Fuchsian groups

Abstract:

The abstract commensurator (aka “virtual automorphisms”) of a group encodes “hidden symmetries”, and is a natural generalization of the automorphism group. In this talk, I will give an introduction to these mysterious and classical groups and then discuss their residual finiteness. Recall that residual finiteness is a property enjoyed by linear groups (by A. I. Malcev), mapping class groups of closed oriented surfaces (by EK Grossman), and branch groups (by definition!). Moreover, by work of Armand Borel, Gregory Margulis, G. D. Mostow, and Gopal Prasad, the abstract commensurator of any irreducible lattice in any “nice enough” semisimple Lie group is locally residually finite (a property is termed “local” if it is satisfied by every finitely generated subgroup of the group). “Nice enough” is sufficiently broad that the only remaining unknown case is PSL_2(R). Are abstract commensurators of lattices in PSL_2(R) locally residually finite? Lattices here are commensurable with either a free group of rank 2 or the fundamental group of an oriented surface of genus 2. I will present a complete answer to this decades old question with a proof that is computer-assisted. Our answer and methods open up new questions and research directions, so graduate students are especially encouraged to attend. This talk covers joint work with Daniel Studenmund.

14:00-14:50 Bob Gilman (Stevens Institute) (Room C 197)

Title: Searching for permutation groups

Abstract: A common way of sampling random subgroups is to choose generators at random from the ambient group. This approach fails for symmetric groups because, as is well known, a random pair of permutations generates the symmetric or alternating group with asymptotic probability 1. However for another method of random choice, a random pair of permutations generates every two-generator permutation group (up to conjugacy) with positive asymptotic probability. This method is based on Kolmogorov complexity and cannot be implemented, but a heuristic variation seems to work. We present some experimental results

15:00-15:50 Jean Pierre Mutanguha (U. of Arkansas) (Room C 197)

Title: Hyperbolic Immersions of Free Groups

Abstract: We proved that the mapping torus of a graph immersion is word-hyperbolic if and only if it has no Baumslag-Solitar subgroups. In this talk, we will illustrate how pullbacks and the Bestvina-Feighn combination theorem were used in the proof.

16:00-16:50 Henry Bradford, (Georg-August Universität Göttingen) (Room C 197)

Title: Short laws for finite groups and residual finiteness growth

Abstract: A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

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Dec 14, 16:00-17:00 Andrzej Zuk (Paris 7) Title: From PDEs to groups