Autumn 2021

Meeting dates: Tuesdays (Topology) 1:50-2:50 pm and Thursdays (Geometric Group Theory) from 10:20 to 11:20 am ET

Seminar organizers - Jingyin Huang, Heejoung Kim, Christoforos Neofytidis, and Rachel Skipper

Autumn 2021 Schedule

The seminars will be run partially in person and partially virtually over Zoom for the Autumn 2021 Semester

https://osu.zoom.us/j/93661626526?pwd=b2xiSEJSTm9BRVlRSitOZXVkMVMzZz09

Zoom ID: 936 6162 6526, Password: 273789

September 23-Heejoung Kim (OSU)-in person

Algorithms detecting stability and Morseness for finitely generated groups

In geometric group theory, finding algorithms for detection and decidability of various properties of groups is a fundamental question. For a finitely generated group G, we can study not only algorithmic problems for G itself but also algorithms related to a particular class of subgroups. For a word-hyperbolic group G, quasiconvex subgroups have been studied widely and there are algorithmic results. For example, Kapovich provided a partial algorithm which, for a finite set S of G, only halts if S generates a quasiconvex subgroup of G. 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, and toral relatively hyperbolic groups.

September 30- Anthony Genevois (Université de Montpellier)

Polycyclic subgroups in asymptotically rigid mapping class groups

The talk will be dedicated to asymptotically rigid mapping class groups associated to punctured planar surfaces obtained by thickening trees, a family of groups that interpolates between braid groups and many Thompson-like groups. After a general introduction and a short survey about what one already knows, I will explain why polycyclic subgroups in such mapping class groups are virtually abelian and undistorted. The central point of the argument is the construction of a diagrammatic representation and a related construction of CAT(0) cube complexes. This is a joint work with A. Lonjou and C. Urech.

October 5- Hannah Hoganson (Utah)-in person

Indicable Subgroups of Big Mapping Class Groups

An important line of inquiry in the study of big mapping class groups is to understand their subgroups. In this talk we will discuss new embedding and combination theorems for indicable subgroups of big mapping class groups. We will also introduce new constructions of indicable groups that can be used as inputs for these theorems. This is joint work with Carolyn Abbott, Marissa Loving, Priyam Patel, and Rachel Skipper.

October 14- Autumn Break - Collin Bleak (St Andrews)

The undecidability of the order problem for Out(G_{n,r}).

We will discuss an interplay between the theory of the automorphism groups of the groups Aut(X_n^Z, sigma_n) of automorphisms of the full two-sided shift on n letters, and the groups Aut(G_{n,r}) of automorphisms of the Higman-Thompson groups G_{n,r}. In particular, using new classifications of both of these groups of automorphisms, we are able to discern that there is a group L_n which is a subgroup of Out(G_{n,r}), and where L_n is isomorphic to the quotient of Aut(X_n^Z, sigma_n) by its center (independent of the value of r for 0<r<n). As a consequence, results in the theory of Aut(X_n^Z, sigma_n) can sometimes be transported directly into the theory of the Higman-Thompson groups. For instance, while the groups G_{n,r} have decidable order problem, the group Out(G_{n,r}) does not. Joint with P. Cameron and F. Olukoya.

October 26- Annette Karrer (Technion)

Contracting boundaries of right-angled Coxeter groups

Associated to a complete CAT(0) space is a topological space called a contracting or Morse boundary. This boundary indicates how similar the CAT(0) space is to a hyperbolic space. Charney-Sultan proved that this boundary is a quasi-isometric invariant, i.e., it can be defined for CAT(0) groups. In this talk we will study contracting boundaries of right-angled Coxeter groups. Right-angled Coxeter groups are CAT(0) groups defined by graphs. In the main part of the talk, we will study when the contracting boundary of a right-angled Coxeter group with totally disconnected contracting boundary remains totally disconnected if we glue certain graphs on its defining graph. This was part of my dissertation. At the end of the talk, we will use our insights to discuss an interesting example where surprising circles appear in the contracting boundary. This was joint work with Marius Graeber, Nir Lazarovich, and Emily Stark.

October 28- Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)

Proper proximality for groups acting on trees

I will discuss joint work with C. Ding where we obtain proper proximality (introduced by Boutonnet, Ioana and Peterson) for many groups acting on trees. Time permitting I will discuss a classification result for graph products of groups.

November 9- Mitul Islam (Heidelberg University)

Convex co-compact groups and relative hyperbolicity

The notion of convex co-compact groups generalizes convex co-compact Kleinian groups from rank one Lie groups to higher rank Lie groups, like PGL_d(R) for d at least three. This generalization encompasses many interesting examples coming from Anosov subgroups and non-Gromov hyperbolic reflection groups. In this talk, we will discuss a geometric property (namely, strongly isolated simplices) that completely characterizes relatively hyperbolic convex co-compact groups (with peripheral subgroups virtually Abelian of rank at least two). This is joint work with Andrew Zimmer.

November 11- Rachel Skipper (OSU)

From simple groups to symmetries of surfaces

We will take a tour through some families of groups of historic importance in combinatorial and geometric group theory, including self-similar groups and Thompson's groups. We will discuss the rich, continually developing theory of these groups which act as symmetries of the Cantor space, and how they can be used to understand the variety of infinite simple groups. Finally, we will discuss how these groups are serving an important role in the newly developing field of big mapping class groups which are used to describe symmetries of surfaces.

November 16- Mehdi Lejmi (CUNY)

Special metrics in Hermitian Geometry

On an almost-Hermitian manifold, the Chern connection connection is the unique connection preserving the almost-Hermitian structure and having J-anti-invariant torsion. It had the property that its (0,1)-part corresponds to the Cauchy-Riemann operator. In this talk, I will discuss the difference between the Chern scalar curvature and the Riemannian scalar curvature induced by the Levi-Civita connection. Then I will discuss the existence of some almost-Hermitian metrics and an analogue of the Yamabe problem for the Chern scalar curvature.

November 18- Sam Shepherd (Vanderbilt University)

A version of omnipotence for virtually special cubulated groups

The theory of group actions on CAT(0) cube complexes is a rich and successful area of research within geometric group theory, particularly regarding the class of virtually special cubulated groups. I will give some background on this and explain my new result for such groups, which allows you to control the orders of the images of certain collections of elements when mapping to a finite group.

November 30- Daniel Woodhouse (University of Oxford)

Regular Cube Complexes and Lieghton's Theorem

I will discuss a large family of homogeneous CAT(0) cube complexes, previously studied by Lazarovich, which offer a natural generalization of regular graphs. I will then show how Leighton's graph covering theorem can be generalized to this setting. More precisely, given such a homogeneous CAT(0) cube complex X, covering two finite cube complexes X_1 and X_2, we will construct a common finite covering of X_1 and X_2. I will discuss potential applications to quasi-isometric rigidity.

December 2-Ilya Kapovich (Hunter College of CUNY)

Primitivity rank for random elements in free groups

Free algebraic structures play a key role in algebra since all other structures can be obtained from them by taking quotients. For example, the polynomial ring $K[x,y]$ (where $K$ is a field) is a free object in the category of commutative algebras over $K$. A free structure always comes with a free basis, although the choice of such a basis is not unique. An element in a free structure is primitive if it belongs to some free basis.

For a finitely generated free group $F=F(X)$ the cardinality of its given free basis $X$ is called the rank of $F$. It is also known that all subgroups of $F$ are themselves free.

In 2014 Doron Puder introduced the notion of primitivity rank $\pi(g)$ for a nontrivial element $g$ in a free group $F_r$ of rank $r$. Namely, $\pi(g)$ is defined as the smallest rank of a subgroup $H$ of $F_r$ containing $g$ as a non-primitive element, or as $\infty$ if not such $H$ exists. The set of all subgroups $H$ of $F_r$ as above is denoted $Crit(g)$. It turned out that primitivity index of an element $w\in F_r$ is closely related to the questions about word-hyperbolicity and subgroup properties of the one-relator group $\langle F_r| w=1\rangle$.

We prove that if $r\ge 2$ and $F_2=F(x_1,\dots, x_r)$ is the free group of rank $r$, then, as $n\to\infty$, for a “random” element $w_n\in F_r$ of length $n$ with probability tending to $1$ one has $\pi(w)=r$ and $Crit(w)=\{F_r\}$. We discuss applications of this result to “word measures” on finite symmetric groups $S_N$, defined by such $w_n$. We also contrast this result with the behavior of the \emph{primitivity index} of random elements in free groups. The latter notion is motivated by studying quantitative aspects of residual finiteness of free groups and by generalizing some results from hyperbolic geometry about “untangling” closed curves on surfaces.