Max Dehn Seminar

on Geometry, Topology, Dynamics, and Groups

Spring 2019 3:15-4:15 LCB 323

Date Speaker Title click for abstract (if available)
August 21
Emily Stark
University of Utah
Cannon--Thurston maps for CAT(0) groups
Two far-reaching methods for studying the geometry of a finitely generated group with non-positive curvature are (1) to study the structure of the boundaries of the group, and (2) to study the structure of its finitely generated subgroups. Cannon--Thurston maps, named after foundational work of Cannon and Thurston in the setting of fibered hyperbolic 3-manifolds, allow one to combine these approaches. Mj (Mitra) generalized work of Cannon and Thurston to prove the existence of Cannon--Thurston maps for normal hyperbolic subgroups of a hyperbolic group. These maps can be used to understand the structure of the boundary of such groups. I will explain why similar theorems fail for certain CAT(0) groups. This is joint work with Benjamin Beeker, Matthew Cordes, Giles Gardam, and Radhika Gupta
September 11
Spencer Dowdall
Vanderbilt University
Discretely shrinking targets in moduli space
Given a nested decreasing family of targets B_n in a measure space X equipped with a flow phi_t (or transformation), the shrinking target problem asks to characterize when there is a full measure set of points x that hit the targets infinitely often in the sense that {n \in N : phi_n(x)\in B_n} is unbounded. This talk will examine the discrete shrinking target problem for the Teichm├╝ller flow on the moduli space of unit-area quadratic differentials and show that for any ergodic probability measure, almost every differential will hit a nested spherical targets infinitely often provided the measures of the targets are not summable. Our key tool is an effective mean ergodic theorem stating that the time-average of any L^2 function converges to its space-average at a uniform rate in L^2. As an application, we obtain a logarithm law describing how quickly generic discrete geodesic trajectories accumulate on a given point. Joint with Grace Work.
September 18
Daniel Woodhouse
Oxford University
Action rigidity of free products of hyperbolic manifold groups
Gromov's program for understanding finitely generated groups up to their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on the same proper geodesic metric space. A *common model geometry* for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is *action rigid* if any group G' that has a common model geometry with G is abstractly commensurable to G. We show that free products of closed hyperbolic manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space. This is joint work with Emily Stark.
September 25
John Smillie
University of Warwick
Dynamics on moduli spaces
A powerful tool for understanding the geometry and dynamics of the torus is to look at dynamics of flows on the moduli space of tori. There are two natural generalisations of this idea. One is to look at higher dimensional tori the other is to flat surfaces of higher genus. In the first case the relevant dynamics are homogeneous dynamics we can apply the powerful results of Ratner and others. The second case involves more exotic dynamics and is more mysterious. I will describe some recent joint work with Jon Chaika and Barak Weiss and some older work with Barak Weiss and explain how it is connected to this question.
October 2
Alexander Rasmussen
Yale University
October 9
No seminar (Fall Break)
October 16
Giovanni Forni
University of Maryland
October 23
Eduard Schesler
Universitaat Bielefeld
The Sigma conjecture for solvable S-arithmetic groups via discrete Morse theory on Euclidean buildings.
Given a finitely generated group G, the Sigma invariants of G consist of geometrically defined subsets Sigma^k(G) of the set S(G) of all characters chi: G -> R of G. These invariants where introduced independently by Bieri-Strebel and Neumann for k=1 and generalized by Bieri-Renz to the general case in the late 80's in order to determine the finiteness properties of all subgroups H of G that contain the commutator subgroup [G,G]. In this talk we determine the Sigma invariants of certain S-arithmetic subgroups of Borelgroups in Chevalley groups. In particular we will determine the finiteness properties of every subgroup G of the group of upper triangular matrices B_n(Z[1/p]) < SL_n(Z[1/p]) that contains the group U_n(Z[1/p]) of unipotent matrices where p is any sufficiently large prime number.
November 6
Genevieve Walsh
Tufts University
November 27
No seminar (Thanksgiving)
January 22
Caglar Uyanik
Yale University

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You may also be interested in the RTG Seminar
Max Dehn Seminar is organized by Mladen Bestvina, Ken Bromberg, Jon Chaika, Osama Khalil,
Priyam Patel, Emily Stark, Domingo Toledo, and Kevin Wortman.

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