Seminar organizers: William Chong, Christopher Karpinski, Piotr Przytycki, Daniel Wise

The 1-skeleton of an ~A_2 Bruhat-Tits building is isomorphic to the Cayley graph of an abstract group with relations coming from ”triangle presentations.” This abstract group either embeds into PGL(3, Fq((x))) or PGL(3, Qq), or else is exotic. Currently, the complete list of triangle presentations is only known for projective planes of orders q=2 or 3. However, one abstract group that embeds into PGL(3,Fq((x))) for any prime power q is known via the trace function corresponding to the finite field of order q^3. I have found a different method to derive this group as well as others via perfect difference sets. This new method demonstrates a previously unknown connection between difference sets and ~A_2 buildings. Moreover, this new method makes the final computation of triangle presentations easier, which is computationally valuable for large q.

The outer automorphism group of the free group Out(F_r) acts as the isometry group on the deformation space of weighted graphs, Culler-Vogtmann Outer space CV_r. The train track theory of Bestvina-Feighn-Handel bridges studying topological representatives of the group elements and geodesics in this space it acts on. We use the asymptotic conjugacy class invariant of the Handel-Mosher ideal Whitehead graph to “stratify” the space of geodesics, and the dynamically minimal “fully irreducible” outer automorphisms, into train track automata for different ideal Whitehead graphs. We then also contextualize this work in the broader program of understanding the geodesic flow. While the flow in the closed hyperbolic manifold and Teichmuller space settings is ergodic, it is unclear whether graphs live in such a nice setting. We explain some of our indicators of certain properties the flow may have. Some results presented are joint with some combinations of Y. Algom-Kfir, D. Gagnier, I. Kapovich, J. Maher, L. Mosher, and S.J. Taylor.

Given an irreducible element of Out(Fn), there is a graph and an irreducible "train track map" on this graph, which induces the outer automorphism on the fundamental group. The stretch factor of an outer automorphism measures the asymptotic growth rate of words in Fn under applications of the automorphism, and appears as the leading eigenvalue of the transition matrix of such an irreducible train track representative. I'll present work showing a lower bound for the stretch factor in terms of the edges in the graph and the number of folds in the fold decomposition of the train track map. Moreover, in certain cases, a notion of the latent symmetry of a graph G gives a lower bound on the number of folds required for any train track map on G. I'll use this to classify all single fold irreducible train track maps.

A major theme of geometric group theory over the past few decades has been to cubulate groups, i.e., construct proper
cocompact actions on CAT(0) cube complexes. Such actions often yield algebraic consequences (e.g. subgroup separability,
biautomaticity, aTmenability), and a group is often obstructed from cubulation by not having one of these algebraic
properties. Taking a different, geometric point of view, we define a class of 'poisonous' spaces -- *richly branching
flats (RBFs)* -- and we show that groups containing RBFs are not quasi-isometric to CAT(0) cube complexes and thus
not cubulated. We present some applications to free-by-cyclic groups and tubular groups. This work is joint with Harry Petyt.

In theory one can find a Morse function on the moduli space M_{g,n} of hyperbolic surfaces of genus g
with n punctures, as Morse functions on a given manifold/orbifold are dense among all smooth functions.
However, it may not be so easy to find a ‘natural’ one.

I will talk about my construction, which improves the known results from topological Morse functions on
M_{g,n}, i.e., the systole function, to C^2-Morse functions (and eventually smooth ones) on the
Deligne-Mumford compactification M_{g,n} bar.

I will start by talking about some background of hyperbolic geometry, Teichmüller theory and Morse theory.

Given an injective endomorphism of a free group, what is the "best" way to represent it so as to read off its dynamical properties? Using Stallings graphs, I'll describe an answer to this question for nonsurjective endomorphisms. To some degree, it turns out nonsurjectivity greatly simplifies matters --- a result that I found rather surprising! I proved that all injective endomorphisms can be uniquely represented by certain kinds of expanding immersions on graphs; a bit paradoxically, this representation is trivial when the endomorphism is an automorphism.

Shephard groups are closely related to complex reflection groups and generalize Coxeter groups and Artin groups. It is well known that Coxeter groups are CAT(0), and it is conjectured that Artin groups are CAT(0). But because their definition is quite general, there are Shephard groups which exhibit seemingly pathological behavior, at least in regards to curvature. We will focus on two such classes. The first is a class of CAT(0) Shephard groups which exhibit “Coxeter-like” behavior, and strictly contains the Coxeter groups. The second class lies more squarely between the Artin and Coxeter groups, and consists of groups which cannot be CAT(0). However, they are relatively and acylindrically hyperbolic. We will give some motivation as to why this behavior occurs and why it doesn’t contradict the conjectural non-positive curvature of Artin groups.

The orbit space of a pseudo-Anosov flow is a topological 2-plane with a pair of transverse (possibly singular) foliations, associated with a well-defined ideal circle introduced by Fenley. Bi-foliated planes were introduced by Barthelmé-Frankel-Mann for describing the orbit spaces of pseudo-Anosov flows, and more recently, Barthelmé-Bonatti-Mann gave a sufficient and necessary condition for reconstructing a bi-foliated plane from its infinity data. From certain circle actions with infinity data, we reconstruct flows and manifolds realizing these actions, including all orientable transitive pseudo-Anosov flows in closed 3-manifolds. This gives a geometric model for such flows and manifolds, applies to a special case of Cannon’s conjecture and gives a description for certain hyperbolic 3-manifolds in terms of the distinct (ordered) triple of the ideal 2-sphere. This work is joint with Hyungryul Baik and Chenxi Wu. A similar result was proved independently by Barthelmé-Fenley-Mann.