# Fall 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 (Room 42414)**

***September 13, ** Heejoung Kim (University of Illinois at Urbana-Champaign)

**Title**: Algorithms detecting stability and Morseness for finitely generated groups

Abstract:

For a word-hyperbolic group *G*, the notion of quasiconvexity is independent on the choice of a generating set and a quasiconvex subgroup of *G* is quasi-isometrically embedded in *G*. In 1994 Kapovich provided a partial algorithm which, on input a finite set *S* of *G*, halts if *S* generates a quasiconvex subgroup of *G* and runs forever otherwise.

However, beyond word-hyperbolic groups, the notion of quasiconveixty is not as useful.

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, a ``stable'' subgroup and a ``Morse'' subgroup.

In this talk, we will discuss various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups.

***September 20, **Susan Hermiller (University of Nebraska)

**Title**: Algorithms for groups of PL homeomorphisms

Abstract:

The group PL_+(I) of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important subgroups, most notably R. Thompson's group F. We study dynamical properties of the homeomorphisms in `computable' subgroups of PL_+(I) (including F) to give an algorithm which determines whether or not any given finite subset of such a computable group generates a solvable group. We give an application of this to solving the subgroup membership problem for a large family of finitely generated solvable subgroups of finitely generated groups of PL homeomorphisms. This is joint work with Collin Bleak and Tara Brough.

***September 27, **

***October 4, **Ariane Masuda and Johann Thiel (New York City College of Technology)

**Title**: Subgroups of SL_{2}(**Z**)$ characterized by certain continued fraction representations

Abstract: For positive integers *u* and *v*, let $L_u=\begin{bmatrix} 1 & 0 \\ u & 1 \end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}$. Let S_{u,v} be the monoid generated by *L*_{u} and *R*_{v}, and *G*_{u,v} be the group generated by *L*_{u} and *R*_{v}. In this talk we will show an extension of a characterization of matrices $M=\begin{bmatrix}a & b \\c & d\end{bmatrix}$ in *S*_{k,k} and *G*_{k,k} when *k* ≥ 2 given by Esbelin and Gutan to *S*_{u,v} when *u,v* ≥ 2 and *G*_{u,v} when *u,v *≥ 3. We will present a simple algorithmic way of determining if *M* is in *G*_{u,v} using a recursive function and the short continued fraction representation of *b*/*d*.

***October 18, **Denis Osin (Vanderbilt University)

**Title: **Quasi-isometric diversity of groups

Abstract: Quasi-isometry is an equivalence relation that identifies metric spaces having the same large scale geometry. It is especially useful in geometric group theory and plays essentially the same role as the isomorphism relation in algebra. It is well-known that the set of quasi-isometry classes of finitely generated groups has the cardinality of the continuum. Indeed, this immediately follows from the existence of continuously many groups with pairwise inequivalent growth functions proved by Grigorchuk in the 80s. A different proof, using small cancellation theory, was given by Bowditch. More recently, continuous families of pairwise non-quasi-isometric groups were constructed inside the classes of solvable groups (Cornulier-Tessera), groups with property FP (Kropholler-Leary-Soroko), and Gromov monsters (Gruber-Sisto). I will explain that all these results can be thought of as particular manifestations of a more general phenomenon, which has its roots in descriptive set theory. We will also discuss applications of this approach to constructing new examples of non-quasi-isometric groups having interesting algebraic and geometric properties.

***October 25, **Piotr Nowak (Polish Academy of Sciences)

**Title: **On property (T) for *Aut(F*_{n}*)*

Abstract:

I will present the proof that *Aut(F*_{n}*)*, the automorphism group of the free group on n generators, has Kazhdan’s property (T) for *n>4*.

The proof uses a characterization of property (T) via an algebraic condition in the group ring due to Ozawa. The strategy involves computer assistance in the form of semidefinite programing (i.e. positive definite convex optimization).

As one of the applications we also obtain new asymptotically optimal estimates of Kazhdan constant for *Aut(F*_{n}*)* and *SL*_{n}*(***Z***)*.

This is joint work with Marek Kaluba and Taka Ozawa (*n=5*) and with Kaluba and Dawid Kielak (*n>5*).

***November 1,** Jean-Camille Birget (Rutgers-Camden)

**Title**: The word problem of the Brin-Thompson groups is coNP-complete

Abstract:

We prove that the word problem of the Brin-Thompson group nV over a finite generating set is coNP-complete (for n ≥ 2). The groups nV with n ≥ 2 are the first examples of finitely presented simple groups with non-easy word problem. Unless NP = coNP, nV (for n ≥ 2) is not embeddable in a finitely presented group with polynomially bounded Dehn function.

The groups nV were introduced by Matt Brin as an n-dimensional generalization of the Thompson group V , which itself was the first known example of an infinite finitely presented, simple group. It was proved by Brin and others that the groups nV form an infinite family of infinite, finitely presented, simple groups.

We also prove that the word problem of the Thompson group V over a certain infinite set of generators, related to boolean circuits, is coNP-complete. The groups nV (and their monoid generalizations) are a bridge between algebra and models of computation.

The talk is based on the preprint arXiv:1902.03852 (Feb. 2019).

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Also happening the week of Oct 28-Nov 1:

***Monday, October 28:** CUNY Graduate Center Math Colloquium, 5pm-6pm, Room 4214 (Math Lounge)

Speaker: Ilya Kapovich (Hunter College)

Title: Counting closed geodesics in the moduli space of Outer space: double exponential growth.

***Friday, November 1: **Special guest lecture, 2pm-3:30pm, Room 3212, Graduate Center

Speaker: Lisa Carbone (Rutgers University)

Title:Simple Lie Groups of Finite and Infinite Dimensions (problems talk for graduate students).

**-----------**

***November 8,** Alfredo Costa (University of Coimbra)

**Title**: The profinite Schützenberger group defined by a symbolic dynamical system.

Abstract:

In finite semigroup theory, free profinite semigroups play a very

important role. Around 2005, Almeida introduced a connection with

symbolic dynamics that proved to be helpful to understand their

structure. One of the most relevant aspects of this connection is the

association between an irreducible symbolic dynamical system X and the

Schützenberger group G(X) of a special regular J-class, defined by X, of

the free profinite semigroup over the alphabet of X.

The profinite group G(X) is a dynamical invariant. In the case of

minimal systems, it has a sort of geometric interpretation: it is the

inverse limit of the profinite completions of the fundamental groups of

the finite Rauzy graphs of X.

In this talk, we survey some of the main results about the group G(X),

ending, if time permits, with a recent application to the theory of

codes (arXiv:1811.03185).

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***November 15,** Alexander Ushakov (Stevens Institute of Technology)

**Title**: Conjugacy problem in the first Grigorchuk's group

Abstract:

We prove that Conjugacy problem in the first

Grigorchuk's group can be solved in linear time.

----------------------------

***November 22,** Robert Gilman (Stevens Institute of Technology)

**Title**: Generic-Case Complexity at Age 16

Abstract:

Geometric group theory has many important computational problems, e.g., the word, conjugacy and isomorphism problems, which are uncomputable. Standard techniques from computer science for analysis of algorithms do not apply to these problems; but generic-case complexity does. In this talk we review the development of generic-case complexity since its introduction in 2003 and speculate on solutions to some of its open problems.

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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