Max Dehn Seminar
on Geometry, Topology, Dynamics, and Groups
Fall 2023
LCB 219
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 (uGibbs/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 nonhyperelliptic strata. More precisely, we show that there exists a generic subset of every nonhyperelliptic 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 EskinMirzakhani and EskinMirzakhaniMohammadi 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 subobjects 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 CullerVogtmann's Outer space CVn is a space of marked metric graphs, and it compactifies to a set of Fntrees. Each Fntree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological Fntree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and LenzhenMasur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus g has dimension 3g4. In this talk, we give the maximal simplex of length measures on an arational Fntree has dimension in the interval [2n7, 2n2]. 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
KatokSpatzier 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 quasiisometry type of such a graph is fully classified by the set of its witness subsurfaces.

January 31 
Zhijing Wendy Wang
Tsinghua University 
TBA

February 7 
Eli Bashwinger
University at Albany 
TBA

February 28 
Jingyin Huang
The Ohio State University 
TBA

March 13 
Barry Minemyer
Bloomsburg University 
TBA

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.