# Geometry, Groups and

Dynamics/GEAR Seminar

The Geometry, Groups and Dynamics/GEAR Seminar, held at the Illinois hub of GEAR, features speakers of interest to network members. The talks are live streamed on the Illinois Math Department Youtube channel (click for playlist). Seminars postings and can also be found on the Illinois Mathematics Department Events Calendar.

Archive of past lectures.

**Current Semester
**

**12:00 pm, Tuesday, March 28, 2017, 243 Altgeld Hall**

Charles Delman (EIU)

Alternating knots and Montesinos knots satisfy the L-space knot conjecture

Abstract: An L-space is a homology \(3\)-sphere whose Heegard-Floer homology has minimal rank; lens spaces are examples (hence the name). Results of Ozsváth - Szabó, Eliashberg -Thurston, and Kazez - Roberts show that a manifold admitting a taut, co-orientable foliation cannot be an L-space. Let us call such a manifold <i>foliar</i>. Ozsváth and Szabó have asked whether or not the converse is true for irreducible \(3\)-manifolds; Juhasz has conjectured that it is. Restricting attention to manifolds obtained by Dehn surgery on knots in \(S^3\), we posit the following: L-space Knot Conjecture. Suppose \( \kappa \subset S^3\) is a knot in the 3-sphere. Then a manifold obtained by Dehn filling along \(\kappa\) is foliar if and only if it is irreducible and not an L-space. Using generalized surface decomposition techniques that build on earlier work of Gabai, Menasco, Oertel, and the authors, we prove that both alternating knots and Montesinos Knots satisfy the L-space Knot Conjecture. We believe these techniques will prove fruitful in the further study of taut foliations in \(3\)-manifolds. Joint work with Rachel Roberts.

**12:00 pm Tuesday, March 14, 2017 in 243 Altgeld Hall**

Mark Sapir (Vanderbilt and Illinois)

Flat submaps in CAT(0) $(p,q)$-maps and maps with angles

Abstract: This is a joint work with A. Olshanskii. Let $p, q$ be positive integers with $1/p+1/q=1/2$. We prove that if a $(p,q)$-map $M$ does not contain flat submaps of radius $\ge r$, then its area does not exceed $c(r+1)n$ where $n$ is the perimeter of $M$ and $c$ is an absolute constant. Earlier Ivanov and Schupp proved an exponential bound in terms of $r$. We prove an estimate similar to Ivanov and Schupp for much more general ``maps with angles" which include, for example, van Kampen diagrams over the presentation of the Baumslag-Solitar group $BS(1,2)$ and many groups corresponding to $S$-machines. We also show that a $(p,q)$ map $M$ tessellating a plane ${\mathbb R}^2$ has path metric quasi-isometric to the Euclidean metric on the plane if and only if $M$ has only finitely many non-flat vertices and faces.

Video

**12:00 pm, Thursday, March 9, 2017, 243 Altgeld Hall**

Mark Bell (Illinois)

Polynomial-time curve reduction

Abstract: A pair of curves on a surface can appear extremely complicated and so it can be difficult to determine properties such as their intersection number. We will discuss a new argument that, when the curve is given by its intersections with the edges of an ideal triangulation, there is always a "reduction" to a simpler configuration in which such calculations are straightforward. This relies on finding an edge flip or a (power of a) Dehn twist that decreases the complexity of a curve by a definite fraction.

Video

**12:00 pm, Thursday, March 2, 2017, 243 Altgeld Hall**

Malik Obeidin (Illinois)

Hyperbolic volumes of random links

Abstract: What does a random link look like? There have been a few different proposed models for sampling from the set of links -- in this talk, I will describe a model based on random link diagrams in the plane. Such diagrams can be sampled uniformly on a computer due to the work of Gilles Schaeffer, so one can experiment with various invariants of links with the topology software SnapPy. I will present data showing what happens with some of the different invariants SnapPy can compute, and I will outline a proof that the hyperbolic volume of the complement of a random alternating link diagram is asymptotically a linear function of the number of crossings. In contrast, for nonalternating links, I will show why the diagrams we get generically represent satellite (and hence nonhyperbolic) links.

Video

**12:00 pm, Tuesday, February 21, 2017, 243 Altgeld Hall**

John P. D'Angelo (Illinois)

Groups Associated with Rational Proper Maps

Abstract: Given a rational proper map $f$ between balls of typically different dimensions, we define a subgroup $\Gamma_f$ of the source automorphism group. We prove that this group is noncompact if and only if $f$ is linear. We show how these groups behave under certain constructions such as juxtaposition and partial tensor products. We then sketch a proof of the following result. If $G$ is an arbitrary finite subgroup of the source automorphism group, then there is a rational map $f$ for which $\Gamma_f = G$. We provide many examples and, if time permits, discuss the degree estimate conjecture. This work is joint with Ming Xiao.

Video

**12:00 pm, Tuesday, February 7, 2017, 243 Altgeld Hall**

Gili Golan (Vanderbilt)

The generation problem in Thompson group F

Abstract: We show that the generation problem in Thompson group F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2-core of subgroups of F, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin and Navas.

Video

**12:00 pm, Thursday, February 2, 2017, 243 Altgeld Hall**

Nick Vlamis (University of Michigan)

Algebraic and topological properties of big mapping class groups

Abstract: There has been a recent surge in studying surfaces of infinite type, i.e. surfaces with infinitely-generated fundamental groups. In this talk, we will focus on their mapping class groups, often called big mapping class groups. In contrast to the finite-type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. I will discuss several such questions and provide some answers. In particular, I will focus on automorphisms of pure mapping class groups and topological generating sets. This work is joint with Priyam Patel.

Video

**12:00 pm, Tuesday, January 24, 2017, 243 Altgeld Hall**

Sarah Mousley (Illinois)

Boundary maps for some hierarchically hyperbolic spaces

Abstract: There are natural embeddings of right-angled Artin groups $G$ into the mapping class group $Mod(S)$ of a surface $S$. The groups $G$ and $Mod(S)$ can each be equipped with a geometric structure called a hierarchically hyperbolic space (HHS) structure, and there is a notion of a boundary for such spaces. In this talk, we will explore the following question: does an embedding $\phi: G \rightarrow Mod(S)$ extend continuously to a boundary map $\partial G \rightarrow \partial Mod(S)$? That is, given two sequences $(g_n)$ and $(h_n)$ in $G$ that limit to the same point in $\partial G$, do $(\phi(g_n))$ and $(\phi(h_n))$ limit to the same point in $\partial Mod(S)$? No background in HHS structures is needed.

Video

**12:00 pm, Thursday, January 19, 2017, 243 Altgeld Hall**

Xinghua Gao (Illinois)

Orders from $\widetilde{PSL_2(\mathbb{R})}$ Representations and Non-examples

Abstract:
Let $M$ be an integer homology 3-sphere. One way to study left-orderability of $\pi_1(M)$ is to construct a non-trivial representation from $\pi_1(M)$ to $\widetilde{PSL_2(\mathbb{R})}$. However this method does not always work. In this talk, I will give examples of non L-space irreducible integer homology 3-spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

Video