Matthew Gentry Durham - UCR Topology Seminar

UCR Topology Seminar

During the Spring 2020 quarter, all of our meetings will be held on Zoom at:

Talks will run from 12-1p PT, but feel free to show up early for socializing with the speaker.

Spring quarter 2020:

April 15, 2020:

Radhika Gupta (Bristol University)
- Uniform exponential growth for groups
  - In this talk, I will start by defining growth of groups and what does it mean for a group to have uniform exponential growth (UEG). I will present a strategy to get UEG for a group which will involve finding a free subgroup using ping-pong lemma. Time permitting, I will talk about a result of Kar and Sageev which states that a group acting freely on a CAT(0) square complex either has UEG or is virtually abelian.

April 22, 2020:

Chris Leininger (UIUC)
- Polygonal billiards, Liouville currents, and rigidity
  - A particle bouncing around inside a Euclidean polygon gives rise to a biinfinite "bounce sequence" (or "cutting sequence") recording the (labeled) sides encountered by the particle. In this talk, I will describe work with Duchin, Erlandsson, and Sadanand, in which we prove that the set of all bounce sequences---the "bounce spectrum"---essentially determines the shape of the polygon. This is consequence of our main result about Liouville currents on surfaces associated to nonpositively curved Euclidean cone metrics. In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of what goes into the proofs.

April 29, 2020:

Jonah Gaster (UW Milwaukee)
- Discrete and continuous harmonic maps between hyperbolic surfaces
  - Combined work of Eels-Sampson and Hartman asserts the existence of a unique smooth harmonic map in the homotopy class of a map of nonzero degree between a pair of compact hyperbolic surfaces. I'll discuss background, present a suitable discretization, and explore conditions on discretizations that ensure convergence of the discrete maps to the smooth harmonic map. This is joint work with Brice Loustau and Léonard Monsaingeon.

May 6, 2020:

Federica Fanoni (CNRS Paris Est)
- Homeomorphic subsurfaces and a good arc graph
  - It is not hard to show that no finite-type surface admits a proper subsurface homeomorphic to the full surface. In this talk I will discuss joint work with Tyrone Ghaswala and Alan McLeay in which we show that, on the contrary, infinite-type surfaces (surfaces whose fundamental group is not finitely generated) admit proper subsurfaces homeomorphic to the full surface. I will then show how this fact can be used to construct an interesting subgraph of the arc graph with a good mapping class group action for a large class of infinite-type surfaces. No previous knowledge of infinite-type surfaces will be required.

May 13, 2020:

Robert Kropholler (Tufts)
- (In)coherence of surface-by-surface groups
  - The class of surface-by-surface groups comes as a natural extension of the class of fibered 3-manifold groups. We are interested in what properties surface-by-surface groups have in common with fibered 3-manifold groups. We are particularly interested in the properties of coherence and algebraic fibering. I will motivate the class of groups we will study and discuss surrounding results. I will then talk about recent work with S. Vidussi and G. Walsh showing that such a bundle is coherent if and only if one of the surfaces is a torus.

May 20, 2020:

Rylee Lyman (Tufts)
- Some new CAT(0) free-by-cyclic groups
  - As with fundamental groups of 3-manifolds fibering over the circle, free-by-cyclic groups form a varied and interesting class of groups whose geometry depends in large part on the corresponding monodromy, in this case an outer automorphism of the free group. For example, Hagen and Wise showed that word-hyperbolic free-by-cyclic groups act virtually cospecially on CAT(0) cube complexes, while Gersten found an example of a free-by-cyclic group that cannot be even a subgroup of a CAT(0) group. Gersten's group admits a cyclic hierarchy, an iterated splitting as a graph of groups with free-by-cyclic vertex groups and cyclic edge groups, terminating in Z times Z. By contrast, we show that a large class of free-by-cyclic groups admitting an additional symmetry act geometrically on CAT(0) 2-complexes. Up to taking powers this includes mapping tori of all polynomially-growing palindromic and symmetric automorphisms. A key tool in the proof is our construction of so-called CTs for free products.

May 27, 2020:

Jonathan Alvaraz (UCR)
- Orderable Groups and the Space of Left-Orderings
  - A left-orderable group is a group which admits a left-invariant total order. In this talk, I'll discuss examples of these groups, some useful lemmas for working with this structure, connections to dynamics, and the topology on the space of left-orders.

June 3, 2020:

Benjamin Russell (UCR)
- Homological Constraints on SCY Groups
  - In dimension 4, every finitely presentable group occurs as the fundamental group of some compact orientable 4-manifold. When studying 4-manifolds via their fundamental groups, one must be content with a more constrained class of spaces. We present an introduction to one such class, that of the symplectic Calabi-Yau manifolds, including the strict homological constraints on their Betti numbers.

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