Max Dehn Seminar
on Geometry, Topology, Dynamics, and Groups
Spring 2019 3:154:15 LCB 323
Date  Speaker  Title click for abstract (if available) 
August 21

Emily Stark
University of Utah 
CannonThurston maps for CAT(0) groups
Two farreaching methods for studying the geometry of a
finitely generated group with nonpositive curvature are
(1) to study the structure of the boundaries of the group,
and (2) to study the structure of its finitely generated subgroups.
CannonThurston maps, named after foundational work of Cannon and Thurston
in the setting of fibered hyperbolic 3manifolds, allow one to combine these
approaches. Mj (Mitra) generalized work of Cannon and Thurston to prove
the existence of CannonThurston 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 Teichmüller flow on the moduli space of unitarea 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 timeaverage of any L^2 function converges to its spaceaverage 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: quasiisometry, 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 torsionfree, Gromov hyperbolic groups that are quasiisometric, 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 
Analogs of the curve graph for infinite type surfaces
The curve graph of a finite type surface is a crucial tool for understanding the algebra and geometry of the corresponding mapping class group. Many of the applications that arise from this relationship rely on the fact that the curve graph is hyperbolic. We will describe actions of mapping class groups of infinite type surfaces on various graphs analogous to the curve graph. In particular, we will discuss the hyperbolicity of these graphs, some of their quasiconvex subgraphs, properties of the corresponding actions, and applications to bounded cohomology.

October 9
 No seminar (Fall Break)  
October 16

Giovanni Forni
University of Maryland 
Twisted translation flows, twisted cohomology and
effective weak mixing
We study cohomological equations and ergodic integrals for
twisted translation flows, define as products of a
translation flow on a translation surface and a linear flow on a circle.
By standard Fourier analysis the questions we consider
reduce respectively to nonhomogeneous cohomology equations
with purely imaginary constant zeroorder term (twisted
cohomological equation) and to ergodic integrals of functions
times an exponential of time with purely imaginary phase
(twisted ergodic integrals). The motivation is twofold:
on the one hand we want to understand a simple example of
3dimensional translation flows, on the other hand
there is a wellknown close connection between twisted ergodic integrals
and spectral measures of translation flows, already exploited in the
work of BufetovSolomyak. In this respect our aim is
to cast their work in more geometric terms and to generalize it.
Our main results results are effective weak mixing results
for translation flows: lower bounds on the dimension of spectral measures
and upper bounds on the speed of weak mixing.

October 23

Eduard Schesler
Universitaat Bielefeld 
The Sigma conjecture for solvable Sarithmetic 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 BieriStrebel and
Neumann for k=1 and generalized by BieriRenz 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 Sarithmetic
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.

October 30

Benjamin Brück
Universität Bielefeld 
Between Tits buildings and free factor complexes
Much of the modern treatment of automorphism groups of free groups is
motivated by analogies with arithmetic groups. I will present a new
family of complexes interpolating between two wellstudied objects
associated to these classes of groups: the free factor complex and the
Tits building of GL_{n}(Q). Each of the new complexes is
associated to the automorphism group Aut(A_{Γ}) of a
rightangled Artin group and has the homotopy type of a wedge of
spheres. The dimension of these spheres forms a new invariant associated
to Aut(A_{Γ}). These complexes can also be seen as an
Aut(A_{Γ})analogue of the curve complex.

November 6

Genevieve Walsh
Tufts University 
Incoherent freebyfree groups
A group G is called coherent if every finitely generated subgroup
of G is finitely presented. We show that freebyfree groups
satisfying a particular homological criterion are incoherent.
This class is large in nature, including many examples of
hyperbolic and nonhyperbolic freebyfree groups.
We apply this criterion to finite index subgroups of
$F_2\rtimes F_n$ to show incoherence of all such groups,
and to other similar classes of groups.
We also discuss some limitations of our methods.
This is joint work with Rob Kropholler.

November 25
JWB 335 Note unusual time and place 
Chenxi Wu
Rutgers University 
TBA
TBA

November 27
 No seminar (Thanksgiving)  
January 22

Caglar Uyanik
Yale University 
TBA
TBA

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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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