We meet Thursdays at 2:50–3:50 pm in Whitney Hall 100E. This semester's organizer is Cary Malkiewich. The seminar has an announcement mailing list open to all.
Topics include: geometric group theory, differential geometry and topology, low-dimensional topology, algebraic topology, and homotopy theory.
Abstract: We show the existence of an (infinitely generated) discrete subgroup of a Lie group (such as SL_n(R)) that has full limit set in its (Furstenberg) boundary and which is not a lattice, we also discuss the possibility of whether this is possible for finitely generated groups. All notions will be explained. Based on work in progress with Subhadip Dey.
Abstract: Regular trees of graphs are inverse limits of particularly simple
inverse systems of finite graphs. They form a 1-dimensional subclass
of the Markov compacta: a class of finitely describable inverse limits
of simplicial complexes, which includes all boundaries of hyperbolic
groups. I will discuss upcoming joint work with Jacek Swiatkowski in
which we use Bowditch's canonical JSJ decomposition to characterize
the 1-ended hyperbolic groups whose boundaries are (regular) trees of
graphs.
Abstract: Topological groupoids describe orbit structures of dynamical systems by capturing their local symmetries. The group of global symmetries, which are pieced together from local ones, is called the topological full group. This construction gives rise to new examples of groups with very interesting properties, solving outstanding open problems in group theory. This talk is about a new connection between groupoids and topological full groups on the one hand and algebraic K-theory spectra and infinite loop spaces on the other hand. Several applications will be discussed. Parts of this connection already feature in work of Szymik and Wahl on the homology of Higman-Thompson groups.
Abstract: In this talk we investigate invariants that count periodic points of a map. Given a self map $f$ of a compact manifold we could detect $n$-periodic points of $f$ by computing the Reidemeister trace of $f^n$ or by computing the equivariant Fuller trace. In 2020 Malkiewich and Ponto showed that the collection of Reidemeister traces of $f^k$ for varying $k|n$ and the equivariant Fuller trace are equivalent as periodic point invariants, and they conjecture that for families of endomorphisms the Fuller trace will be a strictly richer invariant for $n$-periodic points.
In this talk we will explain our new result which confirms Malkiewich and Ponto's conjecture. We do so by proving a new Pontryagin-Thom isomorphism between equivariant parameterized cobordism and the spectrum of sections of a particular parametrized spectrum and using this result to carry out geometric computations.
Abstract: In this talk, we study the algebraic structure of mapping class group Mod(M) of 3-manifolds M that fiber as a circle bundle over a surface. We prove an exact sequence, relate this to the Birman exact sequence, and determine when this sequence splits. We will also discuss the Nielsen realization problem for such manifolds and give a partial answer. This is joint work with Bena Tshishiku and Alina Beaini.
Abstract: Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. One potential application of these functions is towards linearity of the mapping class group, and we will present some partial progress towards understanding these functions for the mapping class group.
Abstract: Given distinct hyperbolic structures m and m' on a closed orientable surface, how many closed curves have m- and m'-length roughly equal to x, as x gets large? Schwartz and Sharp's correlation theorem answers this question. Their explicit asymptotic formula involves a term exp(Mx) and 0<M<1 is the correlation number of the hyperbolic structures m and m'.
In this talk, we will show that the correlation number can decay to zero as we vary m and m', answering a question of Schwartz and Sharp. Then, we discuss extensions of this correlation theorem to the context of higher rank Teichmüller theory and find diverging sequences of SL(3,R)-Hitchin representations along which the correlation number stays uniformly bounded away from zero.
This talk is based on joint work with Xian Dai and joint work in progress with Nyima Kao.
Abstract: Billiards in polygons can exhibit bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry, Teichmuller theory and ergodic theory on homogeneous spaces. The talk will be an introduction to these ideas, aimed at a general mathematical audience.
Abstract: For a vertex $c$ and an integer radius $r$, the sphere $S_r(c)$ is the induced graph on the set of vertices of distance $r$ from $c$. We will show that spheres in the curve graph are typically connected, and discuss connectivity properties of the Gromov boundary. We will also explain the motivation and context for this work, touching tangentially on Cannon's conjecture and convex cocompactness.
Abstract: Suppose we take an arbitrary collection of hyperplanes in n-dimensional Euclidean, hyperbolic, or spherical geometry, along with all of their nonempty intersections. These form a partially ordered set, so we can take the realization and get a topological space, called the Tits complex. One version of the Solomon-Tits theorem says that, if we were to take *all* hyperplanes, the space we get is homotopy equivalent to a wedge of spheres of dimension (n-1).
In this talk I'll describe how to prove a variant of this theorem where we can take just about any reasonable subset of the hyperplanes, and the result still holds. We can furthermore give a presentation of the homology of the resulting space: it has a generator for each polytope cut out by the hyperplanes, and the relations encode subdivision of the polytopes. The proof is quite fun, it's an inductive proof where we add the hyperplanes one at a time and count how many new polytopes, and spheres in the Tits complex, are created. Our main application is to the groups of cut-and-paste operations between these polytopes.