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Max Dehn Seminar

on Geometry, Topology, Dynamics, and Groups

Fall 2023 and Spring 2024
LCB 225
Wednesdays at 3:15 pm

Date Speaker Title click for abstract (if available)
August 23 Davi Obata
Brigham Young University
Understanding the behavior of the orbits of a general dynamical system is a very hard task. Another approach is to try to understand the average behavior of many orbits. This gives the idea of physical measures, measures that capture the statistical behavior of a set of points having positive volume. An important problem in smooth dynamics is to understand mechanisms that imply the existence of physical measures.
In this talk, we will explore the relation between invariant foliations (stable/unstable foliations), certain types of invariant measures (u-Gibbs/SRB), and how this is connected with the problem of finding mechanisms for the existence of physical measures.

September 13 Przemysław Berk
University of Torun
It is an easy observation that the translation flow on an hyperelliptic translation surface is isomorphic to its inverse. We show that this is not the case, when we consider non-hyperelliptic strata. More precisely, we show that there exists a generic subset of every non-hyperelliptic stratum, such that on a translation surface belonging to this subset, the vertical flow is disjoint in the sense of Furstenberg with its inverse.
This is joint work with Krzysztof Frączek and Thierry de la Rue.

September 27 Jon Chaika
University of Utah
For about 2 decades the horocycle flow on strata of translation surfaces was studied, very successfully, in analogy with unipotent flows on homogeneous spaces, which by work of Ratner, Margulis, Dani and many others, have striking rigidity properties. In the past decade Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi proved some analogous rigidity results for SL(2,R) and the full upper triangular subgroup on strata of translation surfaces.

This talk will begin by introducing translation surfaces and describing some of the previously mentioned rigidity before moving onto its goal, that many such rigidity results fail for the horocycle flow on strata of translation surfaces. Time permitting we will also describe some rigidity result for special sub-objects in strata of translations surfaces. This will include joint work with Osama Khalil, John Smillie, Barak Weiss and Florent Ygouf.

October 9 at 2 pm
Unusual day and time
Usual place
Radhika Gupta
Tata Institute
Webb showed that the arc complex of a surface of high enough complexity does not satisfy a combinatorial isoperimetric inequality: that is, for every N, there is a loop of length 4 in the arc complex that only bounds discs containing at least N triangles. He showed that the same is true for the free splitting complex and the cyclic splitting complex associated with Out(Fn) and concludes that these complexes do not admit a CAT(0) metric with finitely many shapes. On the contrary, he proved that the curve complex satisfies a linear combinatorial isoperimetric inequality. In this talk, we will show that the free factor complex associated with Out(Fn) also fails to satisfy a combinatorial isoperimetric inequality.

October 18 Jacob Russell
Rice University
The theory of convex cocompact subgroups of the mapping class group contains two intertwining threads. One is the rich analogy between these subgroups of the mapping class group and the convex cocompact Kleinian group that inspired them. The other is work of Farb and Mosher plus Hamenstädt that shows convex cocompactness is precisely the property that characterizes when an extension of a surface group is Gromov hyperbolic. Both of these threads have natural generalizations that are unresolved. Among Kleinian groups, convex cocompactness is a special case of geometric finiteness, yet no robust notion of geometric finiteness has emerged for the mapping class group. In geometric group theory, there are a variety of generalizations of Gromov hyperbolicity, but there is no characterization of these geometries for surface group extensions. Mosher suggested these two threads should continue to intertwine with geometric finiteness in the mapping class group (however it is eventually defined) being equivalent to some generalization of Gromov hyperbolicity of the corresponding surface group extension. Inspired by their work on Veech groups, Dowdall, Durham, Leininger, and Sisto conjectured that this generalized hyperbolicity could be the hierarchical hyperbolicity of Behrstock, Hagen, and Sisto. We provide evidence for this conjecture by showing that several classes of subgroups that should be considered geometrically finite (stabilizers of multicurves, twist subgroups, cyclic subgroups) correspond to surface group extensions that are hierarchically hyperbolic.

November 8 Thomas Hill
University of Utah
The work of Mann and Rafi gives a classification surfaces Σ when Map(Σ) is globally CB, locally CB, and CB generated under the technical assumption of tameness. We restrict our study to the pure mapping class group and give a complete classification without additional assumptions. In stark contrast with the rich class of examples of Mann–Rafi, we prove that PMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface, and locally CB or CB generated if and only if Σ has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures.

November 15 Sanghoon Kwak
University of Utah
The Culler--Vogtmann's Outer space CVn is a space of marked metric graphs, and it compactifies to a set of Fn-trees. Each Fn-tree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological Fn-tree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and Lenzhen--Masur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus g has dimension 3g-4. In this talk, we give the maximal simplex of length measures on an arational Fn-tree has dimension in the interval [2n-7, 2n-2]. This is a joint work with Mladen Bestvina, Jon Chaika, and Elizabeth Field.

November 22 No seminar, Thanksgiving
November 29 Homin Lee
Northwestern University
We will discuss about smooth actions on manifolds by higher rank lattices, such as lattices in SL(n,R) with n at least 3. The higher rank property of the acting group suggests that the actions are rigid, which means that the action should have an algebraic origin, such as the Zimmer program and the Katok-Spatzier conjecture. One of the main topics is about how we can provide an algebraic structure on the acting space which is a smooth manifold. We survey some of recent breakthroughs and then we focus on actions on manifolds with “positive entropy” by lattices in SL(n,R), with n at least 3. When the manifold has dimension n, then we will see that the lattice is commensurable to SL(n,Z) from certain "algebraic structure" on M coming from the dynamics. It also gives certain corollaries about the topological properties of M. Part of the talk is ongoing work with Aaron Brown.

December 6 Michael Kopreski
University of Utah
A multiarc and curve graph is a simplicial graph whose vertices are arc and curve systems on a compact, connected, orientable surface S. We show that all multiarc and curve graphs preserved by the natural action of PMod(S) and whose adjacent vertices have bounded geometric intersection number are hierarchically hyperbolic spaces with respect to witness subsurface projection. In addition, we prove that the PMod(S)-equivariant quasi-isometry type of such a graph is fully classified by the set of its witness subsurfaces.

January 12 4-5 pm
LCB 222
Unusual day, time and place
Nicholas Miller
University of Oklahoma
Geodesic submanifolds have played an important role in the study of hyperbolic manifolds, especially in the construction of manifolds in dimension at least 4. In this talk, we will first survey what is known about manifolds in the higher dimensional setting and their relationship to geodesic submanifolds. We will then go on to recount work of Ratner, which connects the study of these submanifolds to that of invariant measures on homogeneous spaces. Finally, we will show how these measures can be important tools in the construction of invariant measures on certain inhomogeneous spaces, which have played an important role in recent rigidity theorems.

January 19 4-5 pm
LCB 222
Unusual day, time and place
Francisco Arana Herrera
University of Maryland
Cutting a hyperbolic surface along a simple closed geodesic yields a hyperbolic subsurface with totally geodesic boundary. For a given hyperbolic surface, what is the expected geometry of these subsurfaces? We discuss recent joint work with Aaron Calderon showing that these subsurfaces look like random metric ribbon graphs in the sense of Kontsevich. In particular, the limiting law does not depend on the initial choice of hyperbolic surface. This result strengthens Mirzakhani’s famous counting results for simple closed geodesics on hyperbolic surfaces.

January 24 Rachel Skipper
University of Utah
For an automorphism of a totally disconnected, locally compact (tdlc) group, Willis introduced the notion of scale which arose in the development of the general theory of these groups. In this talk, we will discuss the setting where the tdlc group is Neretin's group and where the automorphism comes from conjugation in the group. This is an ongoing joint work with Michal Ferov and George Willis at the University of Newcastle.

January 31 Zhijing Wendy Wang
Tsinghua University
In this talk we discuss rigidity and classification of the smooth centralizer of a perturbation of an affine map on a homogeneous space. We mainly focus on the left multiplication of a generic diagonal matrix on SL_n R/\Gamma, where \Gamma is a cocompact lattice. We show that in this case, the smooth centralizer of a perturbation is either a Lie group of dimension zero or 1, or virtually R^{n-1}, and in the latter case, the perturbation is smoothly conjugate to a diagonal matrix acting by left multiplication.

February 7 Eli Bashwinger
University at Albany
In this talk, I will discuss some results concerning the von Neumann algebras of Thompson-like groups arising from $d$-ary cloning systems (for $d \ge 2$). This talk will essentially be a survey of some of the many results coming from three papers I have written so far on this topic, the first two of which I wrote with Matthew Zaremsky, who invented cloning systems with Stefan Witzel in 2018. Given a $d$-ary cloning system on a sequence of groups $(G_n)_{n \in \mathbb{N}}$, we can form a Thompson-like group denoted by $\mathscr{T}_d(G_*)$, and this group canonically contains $F_d$, the smallest of the Higman-Thompson groups. The group inclusion $F_d \le \mathscr{T}_d(G_*)$ translates to an inclusion of their group von Neumann algebras $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$. We have several results concerning this inclusion as well as some results concerning the von Neumann algebra $L(\mathscr{T}_d(G_*))$ itself, some of which will have group-theoretic consequences for these Thompson-like groups and how $F_d$ sits inside them.

February 28 Jingyin Huang
The Ohio State University
The K(pi,1)-conjecture for reflection arrangement complements, due to Arnold, Brieskorn, Pham, and Thom, predicts that certain complexified hyperplane complements associated to infinite reflection groups are Eilenberg-MacLane spaces. We establish a close connection between simple properties about 4-cycles in metric graph theory and this conjecture, via elements of non-positively curvature geometry. We also propose a new approach for studying this conjecture. As a consequence, we deduce a large number of new cases of Artin groups which satisfy the K(pi,1)-conjecture. If time allows, I will also discuss some connections between the K(pi,1)-conjecture and the study of configurations of arcs on surfaces, as well as solving equations in free groups.

March 6 No seminar Spring break
March 13 Barry Minemyer
Bloomsburg University
In 1987 Gromov and Thurston developed the first Riemannian manifolds that are not homotopy equivalent to a hyperbolic manifold but admit a Riemannian metric that is ϵ-pinched for any given ϵ>0. The manifolds that they construct are branched covers of hyperbolic manifolds, and to construct the metric they perform a sort of "geometric surgery" about the ramification locus. In 2022 Stover and Toledo proved the existence of similar branched cover manifolds built out of complex hyperbolic manifolds, and via a result of Zheng these manifolds admit a negatively curved Kahler metric. In this talk we will discuss how to construct a (not Kahler) Riemanain metric on these Stover-Toledo manifolds which is ϵ-close to being 1/4-pinched for any prescribed ϵ>0. These provide the first known examples of Kahler manifolds that are not homotopy equivalent to a complex hyperbolic manifold but admit a Riemannian metric that is ϵ-close to being 1/4-pinched.

April 3 Juhun Baik
The Cantor set arises naturally in various dynamical systems. In particular, the iteration of a complex polynomial yields an invariant set (Julia set), which is often a Cantor set. In this talk, I will focus on the iteration of quadratic polynomials and how the Julia set changes when the parameter varies. The behavior of the Julia set leads to a mapping class of a plane-Cantor set as a result. I will also discuss how the mapping class is constructed. This is a joint work with my advisor Prof. Harry Hyungryul Baik.

April 10 Carlos Ospina
University of Utah
In this talk we will define the Rel foliation for a stratum of translation surfaces with at least two singularities. We will focus on the real Rel flow in the stratum $\mathcal{H}(1,1)$. We will provide some examples of orbits, and their closures. Finally, we will describe the real Rel orbits of tremors of surfaces and provide explicit examples of trajectories that are not recurrent, but do not diverge.

May 2 at 3 pm
LCB 323
Unusual day, time and place
Dick Canary
University of Michigan
Francois Labourie introduced the theory of Anosov subgroups of semi-simple Lie groups in his seminal work on Hitchin representations. Anosov groups are now widely recognized as the higher rank generalization of convex cocompact subgroups of rank one Lie groups. (For our purposes, all higher rank Lie groups will be SL(d,R) and rank one Lie groups will all be PSL(2,R), PSL(2,C) or SO(d,1).) We will review classical results on convex cocompact subgroups of rank one Lie groups and discuss how they generalize into the setting of Anosov groups. Heckling will be encouraged, and may be required, depending on the mood of the speaker.

Archive of past talks         

You may also be interested in the RTG Seminar
Max Dehn Seminar is organized by Mladen Bestvina, Ken Bromberg, Jon Chaika, Elizabeth Field,
Priyam Patel, Rachel Skipper, Domingo Toledo, Kurt Vinhage and Kevin Wortman.

This web page is maintained by Jon Chaika.