# 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 Teichmüller 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 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 non-homogeneous cohomology equations with purely imaginary constant zero-order 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 two-fold: on the one hand we want to understand a simple example of 3-dimensional translation flows, on the other hand there is a well-known close connection between twisted ergodic integrals and spectral measures of translation flows, already exploited in the work of Bufetov-Solomyak. 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 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. 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 well-studied objects associated to these classes of groups: the free factor complex and the Tits building of GLn(Q). Each of the new complexes is associated to the automorphism group Aut(AΓ) of a right-angled Artin group and has the homotopy type of a wedge of spheres. The dimension of these spheres forms a new invariant associated to Aut(AΓ). These complexes can also be seen as an Aut(AΓ)-analogue of the curve complex. November 6 Genevieve Walsh Tufts University Incoherent free-by-free groups A group G is called coherent if every finitely generated subgroup of G is finitely presented. We show that free-by-free groups satisfying a particular homological criterion are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free 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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