# Spring 2019

**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.**

***January 14, 1:00 pm **

### Dominik Francoeur (U of Geneva)

On free subsemigroups in automata semigroups

Groups and semigroups generated by automata can possess many striking properties. For instance, one can find among them groups of intermediate growth and finitely generated infinite torsion groups. One can wonder what can be deduced about the algebraic properties of an automaton group or semigroup from the combinatorial properties of the automaton that generates it. In this talk, we will discuss one such connection, namely the link between the reversibility of the automaton and the existence of a free subsemigroup. This is joint work with Ivan Mitrofanov.

***February 1 Ilya Kapovich (Grad Center and Hunter College CUNY) **

**Title**: The primitivity index function for a free group, and untangling closed geodesics on hyperbolic surfaces.

Abstract: An important result of Scott from 1980s shows that every closed geodesic on a compact hyperbolic surface can be lifted (or ``untangled") to a simple closed geodesic in some finite cover of that surface. Recent work of Patel and others initiated quantitative study of Scott's result, which involves understanding the smallest degree of a cover where a closed geodesic ``untangles", compared with the length of the curve.

In a free group F an element x of F is called ``primitive'' if x belongs to some free basis X of F. In the free group context, primitive elements are algebraic counterparts of simple closed curves on surfaces.

Motivated by these results of Scott and Patel, we introduce several ``untangling'' indexes for nontrivial elements of a finite rank free group F, such as the ``primitivity index'', the ``simplicity index'' and the ``non-filling index". We obtain several results about the worst-case behavior of the corresponding index functions

and about the probabilistic behavior of the indexes on ``random'' elements of F.

We also discuss applications of these results to the original setting of Scott and Patel of untangling closed geodesics on hyperbolic surfaces.

The talk is based on a joint paper with Neha Gupta, with an appendix by Khalid Bou-Rabee.

Pdf slides of the talk

***February 8**: No seminar

*** February 15: **Caglar Uyanik (Yale University)

** **

** Title:** Atoroidal dynamics of subgroups of Out(*F*_{N})

Abstract: I will discuss several examples to illustrate how the dynamics of the Out(F_{N}) action on various spaces reflects on the algebraic structure of the Out(*F*_{N}) itself. More precisely, I will talk about a new subgroup classification theorem for

Out(*F*_{N}) (joint with Matt Clay).

* **February 22: **Jennifer Taback (Bowdoin College)

** Title: **Growth and medium-scale curvature in solvable Baumslag-Solitar groups

Abstract: I will describe a procedure to produce a geodesic representative for

every element of* BS(1,n) = <t,a | tat*^{-1}* = a*^{n}*>*, using a

lattice-based approach in **Z**^{∞}. This yields a straightforward

formula for word length which allows us to prove that there are sets of

elements of positive density of positive, negative and zero medium-scale

curvature, as defined by Bar Natan, Duchin and Kropholler. Moreover, we are

able to show that there are sets of elements of positive density with zero and

negative medium-scale curvature, respectively, when this curvature is computed

over balls of radius *r* for *r>1*. A subset of these geodesic

representatives forms a regular language, and we show that the growth rate of

this language is the growth rate of the entire group.

* **March 1**, Elizabeth Field (University of Illinois at Urbana-Champaign)

**Title:** Trees, dendrites, and the Cannon-Thurston map

Abstract:

When *1 → H → G → Q → 1* is a short exact sequence of three word-hyperbolic groups, Mahan Mj has shown that the inclusion map *i: H → G* extends continuously to a map *∂i: ∂H → ∂G* between the Gromov boundaries of *H* and *G*. This boundary map is known as the *Cannon-Thurston map*. In this context, Mitra associates to every point *z ∈ ∂Q *an ``ending lamination'', *Λ*_{z}* ⊆ ∂*^{2}*H* on *H*. We prove that for each *z ∈ ∂Q*, the quotient space *∂H*/*Λ*_{z}* * is a dendrite, that is a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who proved that in the case where *H=F*_{N} and *Q* is a convex cocompact purely atoroidal subgroup of Out(*F*_{N}) one can identify *∂H*/*Λ*_{z}* *with a certain **R**-tree *T*_{z} in the boundary of the Outer space CV_{N}.

* **March 8**, Alexei G. Miasnikov (Stevens Institute of Technology)

**Title:** Full rank presentations and random nilpotent groups

Abstract:

We study finitely generated nilpotent groups of class c > 1 given by finite presentations of full rank (the matrix of relations in the abelianization of the group has full rank). The class of such groups is rather large since an arbitrary finite presentation (not necessary in the variety of nilpotent groups) asymptotically almost surely has full rank. This gives one a powerful tool to approach random nilpotent groups in the few relations model. We will discuss some quite unexpected properties of random nilpotent groups.

The talk is based on joint results with Albert Garreta and Denis Ovchinnikov.

* **March 15**, Igor Rivin (Temple University)

**Title:** Computational experiments, and how to make sense of them

Abstract: We describe some (mostly group theoretic) experiments, and try to see what modern data analysis methods tell us about the world.

* **March 22: **Behrang Forghani (Bowdoin College)

** Title: **Boundaries and Entropy of Groups

Abstract: Kaimanovich's *ray* and *strip* *approximations* are the primary tools to identify Poisson boundaries of random walks on discrete groups. In this talk, I will outline how to extend these geometric criteria to the random walks on *locally compact groups*. In order to do that, a version of *Shannon-McMillan-Breiman * theorem for locally compact groups will be established. In the end, I will discuss some applications to identify the Poisson boundary of locally compact groups which act by isometries on non-positively curved spaces. This is joint work with Giulio Tiozzo.

* **March 29: No seminar**

* **April 5**, Alina Vdovina (Newcastle University, U.K.)

**Title: **Ramanujan cube complexes and non-residually finite CAT(0) groups in any dimension

Abstract: We will construct groups acting cocompactly on products of n>2 trees using quaternion algebras and show that their quotients provide first explicit examples of Ramanujan cube complexes in any dimension. We will describe so-called doubling construction, which allows to get cube complexes in any dimensions with non-residually finite fundamental groups

**Note:** Alina Vdovina will also be giving a GRECS Seminar talk "Higher-dimensional expanders with applications to clustering" at the Hunter College of CUNY, Wednesday, April 3, room 921 Hunter East

* **April 12**, Leonid Potyagailo (University of Lille, and CRM (Montreal))

**Title:** Martin and Floyd boundaries of finitely generated groups

Abstract:

The talk is based on two recent preprints :

1. [GGPY], I. Gekhtman, V. Gerasimov, L.P. W. Yang, "Martin boundary covers Floyd boundary"

(arXiv:1708.02133),

2. [DGGP], M. Dussaule, I. Gekhtman, V. Gerasimov, L.P., "The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups" (arXiv:1711.11307).

We study two different compactifications of finitely generated groups. The first is the Martin compactification which comes from the random walks on the Cayley graph of a group equipped with a symmetric probability measure. The second compactification is the Floyd compactification which is the Cauchy completion of the Cayley graph equipped with a distance obtained by a rescaling of the word metric.

The corresponding boundaries are the remainders of the group in these compactifications.

Our first main result from [GGPY] states that the identity map on the group extends to an equivariant and continuous map between Martin and Floyd compactifications. The proof is based on our generalization of the Ancona inequality proved by A. Ancona for hyperbolic groups in 80's.

Using these results we prove in [DGGP] that the Martin boundary of a hyperbolic group $G$ relatively to a system of virtually abelian subgroups is a "parabolic blow-up space". It is obtained from the limit set $X$ of the relatively hyperbolic action of $G$ by replacing every parabolic fixed point $p\in X$ by the euclidean sphere of dimension $k-1$ where $k$ is the rank of its parabolic stabilizer. All other points of $X$ are conical and they remain unchanged.

* May 3, Vincent Guirardel (Rennes)

**Title:** Towers, elementary equivalence and elementary embeddings for hyperbolic groups.

Abstract:

A group is* elementarily free* if it cannot be distinguished from a free group from the point of view

of first order logic.

As a consequence of Sela's and Kharlampovich-Miasnikov's solution of Tarski's problem,

the simplest examples of elementarily free groups are free products of free groups and (non-exceptional) surface groups.

In particular, finite index subgroups of such groups are still elementarily free, and

one can wonder how general is this phenomenon.

We prove that in fact, these are the only examples: any finitely generated elementarily free group which is not

a free product of free groups and surface groups has a finite index subgroup which is not elementarily free.

The key concept involved is the notion of tower (a.k.a NTQ-groups).

Several variants of this notion have been introduced by various authors,

and can be used to characterize elementary equivalence

and elementary embeddings among torsion-free hyperbolic groups.

This is a joint work with Gilbert Levitt and Rizos Sklinos

* May** 10**, Olga Kharlampovich and Laura Lopez (CUNY Graduate Center)

**Title:** Diophantine problem in some metabelian groups.

Abstract:

In free metabelian non-abelian groups equations are undecidable (Roman'kov). In fact, in a finitely generated metabelian group *G* given by a finite presentation in the variety of metabelian groups the Diophantine problem is undecidable asymptotically almost surely if the deficiency of the presentation is at least 2 (Garreta, Miasnikov, Ovchinnikov). In general, if the quotient of a finitely generated metabelian group *G* by its third term of the lower central series is a non-virtually abelian nilpotent group, then the decidability of the Diophantine problem in *G* would imply decidability of the Diophantine problem for some finitely generated ring of algebraic integers associated with this quotient. The latter seems unlikely, since there is a well-known conjecture in number theory that states that the Diophantine problem in rings of algebraic integers is undecidable. The discussion above shows that finitely generated metabelian groups *G* with virtually abelian quotients by the third term of the lower central series present an especially interesting case in the study of equations in metabelian groups.

Metabelian Baumslag-Solitar groups and wreath products **Z**_{n}* wr ** Z* and

**Z***wr*

*, where*

**Z***is an infinite cyclic group and*

**Z**

**Z**_{n}is a cyclic group of order

*n*, are the typical examples of such groups. We will prove decidability of the Diophantine problem in these groups and discuss their other interesting properties. These are joint results with A. Miasnikov.

* May** 17**, Rizos Sklinos (Stevens Institute of Technology)

**Title:** Nonequational stable groups

Abstract:

Equationality resembles a notion of Notherianity in the abstract

setting of model theory. It is a notion strictly stronger than

stability, but it was thought for many years that all natural stable

structures are equational. In 2006 Sela proved that the theory of the

free group is not equational but stable, making it the first natural

example of a stable nonequational theory. His proof used the

complicated machinery developed for answering Tarski's question (answered by Kharlampovich-Myasnikov as well). In

this talk I will present an elementary trasnparent proof for the

nonequationality of the free group. As a matter of fact this proof

extends to any free product of groups (except **Z**_{2}*****Z**_{2}). This is

joint work with Isabel Müller.

The seminar meets Friday 4:00-5:00 p.m. at the Graduate Center of the City University of New York (Room 5417).

The current organizers are:

Robert Gilman (Stevens Institute of Technology), rhgilman@gmail.com

Ilya Kapovich (Hunter College of CUNY), ikapovitch@gmail.com

Olga Kharlampovich (Hunter College of CUNY), okharlampovich@gmail.com,

Alexei Miasnikov (Stevens Institute of Technology), amiasnikov@gmail.com

Vladimir Shpilrain (City College of CUNY), shpilrain@yahoo.com

Benjamin Steinberg (City College of CUNY), bsteinberg@ccny.cuny.edu

If you would like to give a talk, or have a suggestion for a seminar speaker, please e-mail one of the organizers. If you want to be added to/removed from the NYGT Seminar mailing list, please e-mail Ilya Kapovich at ikapovitch@gmail.com.

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