Categories
Mini-sessions

Geometric group theory: Non-positively curved groups

Actions of big mapping class groups on the arc graph- Carolyn Abbott (Columbia)

Abstract: Given a finite-type surface (i.e. one with finitely generated fundamental group), there are two important objects naturally associated to it: a group, called the mapping class group, and an infinite-diameter hyperbolic graph, called the curve graph. The mapping class group acts by isometries on the curve graph, and this action has been extremely useful in understanding the algebraic and geometric properties of mapping class groups. One particularly important class of elements of the mapping class group are those which act as loxodromic isometries of the curve graph; these are called “pseudo Anosov” elements. Given an infinite-type surface with an isolated puncture, one can associate two analogous objects: the big mapping class group and the (relative) arc graph. In this talk, we will consider the action of big mapping class groups on the arc graph, and, in particular, we will construct an infinite family of “infinite-type” elements that act as loxodromic isometries of the arc graph, where an infinite-type element is (roughly) one which is not supported on any finite-type subsurface. This is joint work with Nick Miller and Priyam Patel.

Video

 

Morse Quasiflats- Jingyin Huang (Ohio State)

Abstract: We introduce the notion of Morse quasiflats, which is a common generalization of Morse quasigeodesics and quasiflats of top-rank. In the talk, we will provide motivations and examples for Morse quasiflats, as well as a number of equivalent definitions and quasi-isometric invariance (under mild assumptions). We will also show that Morse quasiflats are asymptotically conical, and provide several first applications. Based on joint work with B. Kleiner and S. Stadler.

Video

Part 1/2

Part 2/2

 

Effective ping-pong in CAT(0) cube complexes- Kasia Jankiewicz (University of Chicago)

Abstract: Ping-pong lemma is a useful tool for finding elements in a given group that generate free semigroups. We use it to prove uniform exponential growth of certain groups acting on CAT(0) cube complexes. We also construct examples of groups with arbitrarily large gap between their cohomological and cubical dimensions. This is partially joint work with Radhika Gupta and Thomas Ng.

Video

 

Mod p and torsion homology growth in nonpositive curvature- Kevin Schreve (University of Chicago)

Abstract: We compute the growth of mod p homology in finite index normal subgroups of right-angled Artin groups. We give examples where it differs from the rational homology growth, find such a group with exponential torsion growth, and for odd primes p construct closed locally CAT(0) manifolds with nontrivial mod p homology growth outside the middle dimension.

Video

 

11 replies on “Geometric group theory: Non-positively curved groups”

Hi Kasia, thanks for explaining the interesting difference between proper actions and free proper actions. The following question is not very related to your main results, but I’ll ask anyways since it is kind of within the topic.

If G is the fundamental group of some compact *special* cube complex X, there are potentially different choices of such X with pi_1(X)=G. Is there a way to give an upper bound of the smallest possible dimension of X in terms of some other notions of dimension related to G? Maybe assume G to be hyperbolic if that helps.

Hi Joe, Sorry I missed your comment earlier. The C'(1/6) small cancellation groups are hyperbolic and cocompactly cubulated, so they are also virtually special. In the examples I construct the smallest dimension of a cube complex can be arbitrarily far from 2 (which is the cohomological dimension, as well as the CAT(0) dimension of these groups). However, my result does not say anything about finite index subgroups of these groups, so in particular, I don’t know anything about the smallest dimension of a cube complex corresponding to a finite index subgroup that is special. So I don’t really have an answer, but I think it’s a very interesting question.

Hi, my name is Jingyin Huang and I am currently a postdoc at Ohio State University. My current research interests lies around the asymptotic geometry and combinatorics of groups and spaces with non-positively curved feature.

I will be happy to answer questions about my talk through the comment section. If you prefer to ask question face to face, then my Zoom office hours will be 10 am EST on June 8 (the second week of NCNGT).

Hello, I’m Kevin Schreve and I am a postdoc at UChicago. I like to study connections between aspherical manifolds and geometric group theory.

I am very happy to answer questions about my talk, see you in the comments!

Hi, everyone! I’m Carolyn Abbott and currently a postdoc at Columbia University. I generally study isometric actions of groups on hyperbolic spaces, often under additional assumptions on the action, such as acylindricity or the existence of WPD elements.

I’m happy to answer any questions you may have — I look forward to exchanging comments with you! I also have Zoom office hours on Tuesday, June 2, at 1pm (EST). Feel free to stop by if you’d like to talk face-to-face.

Hi Carolyn, thanks for your great talk! This is a nice and flexible construction once we have the subsurface Sigma’. You mentioned at the end that some simple closed curve will converge to an invariant lamination. Do you expect all (essential) simple closed curves behave the same way? Maybe a related question is: what is the dynamics of your pseudo-Anosov element on the circle of rays coming out of the puncture? Is there exactly *one* pair of sink and source?

Hi Joe, Thank you! And good questions! For laminations, the situation is definitely different here than for pseudo-Anosovs in the finite-type setting: there can be simple closed curves that are fixed by our homeomorphisms (take anything on the “back” of Sigma’), and simple closed curves that limit to the empty set (consider the ladder surface with the handles pulled out, and take a scc on one of the handles — iterating our homeomorphisms just make this scc “march out to infinity”). But even given those differences, yes, I do think it is likely true that every scc limits to an invariant lamination, though I don’t yet know how to prove it. As for the circle of rays, this is a great question that I have not thought about at all. My first instinct is that there probably is exactly one pair of sink and source, but I’m really not sure. Let me think about this a bit more and then I’ll come back and respond again.

Hi Carolyn, thanks for pointing out that lots of simple closed curves are invariant, which I didn’t notice earlier… A naive follow-up: Could a loxodromic mapping class acting on the arc complex of a closed surface with one marked point also preserve some simple closed curve? Are such elements necessarily pseudo-Anosov? I only know that loxodromic<->pseudo-Anosov for closed surfaces and the curve complex.

Comments are closed.