Seminar organizers: William Chong, Christopher Karpinski, Piotr Przytycki, Daniel Wise
In classical harmonic analysis, Poisson and Dirichlet studied what form do harmonic functions on the unit ball of R^d take. The 20th century reworked this question in the setting of groups. Furstenberg showed that for every (locally compact second countable) topological group G and a Borel probability measure \mu on G, there is a measure space (B,\nu), called the Poisson boundary -- in analogy to the boundary of the unit ball --, which gives a natural representation for all \mu-harmonic functions on G. I will review and explain the foundational results and then talk about joint work with Yair Hartman and Omer Segev in which we give conditions for probability measures \mu_1 and \mu_2 to share the Poisson boundary.
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.
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.
I will give a quick survey of the known results and methods towards the Kervaire conjecture in combinatorial group theory. Then i will offer a small but pretty result that offers a new paradigm. This is joint work with Andy Ramirez-Côté.
An acylindrical action generalizes proper and cobounded actions on hyperbolic spaces. Non-elementary acylindrical actions provide acylindrically hyperbolic groups, which includes most mapping class groups of punctured surfaces, 3-manifold groups, and Out($F_n$) for n > 1. In this talk, we will explore how acylindricity of a group action on a tree can be preserved under quotients by certain subgroups, and discuss the existence of a largest acylindrical action for some groups acting on trees. In addition, we will show when Out(BS(p,q)) is acylindrically hyperbolic for non-solvable Baumslag-Solitar groups, despite BS(p,q) itself not being acylindrically hyperbolic, and explore further applications of these acylindricity results. This is a joint work with Daxun Wang.
For any partial HNN extension H, we describe a construction of an embedding of H to an ascending HNN extension G. We use disk diagrams to also show that G is relatively hyperbolic to H. Basic definitions of small-cancellation theory and relative hyperbolicity will be covered in the talk. This is a joint work with Daniel Wise.
The cactus group has been studied in various settings, first appearing as a "mock reflection group". Its “pure” subgroup is the fundamental group of the iterative blowup of a hyperplane arrangement (the Deligne-Mumford space). The virtual cactus group is a modified version of the cactus group. In recent years, techniques from geometric group theory were used to determine properties of the cactus group. It was shown that the cactus group is virtually special compact. In this talk, we will present this viewpoint and prove that the virtual cactus group is also virtually special compact.
At the dawn of the millenium, Whyte completed the quasi-isometry classification of the Baumslag-Solitar groups. The analogous problem of their measure equivalence classification, however, has only seen partial progress. Of note, a decade ago, Kida proved that under certain mixing conditions, no measure equivalence coupling can exist between some of these groups, e.g. BS(2,3) and BS(5,7). In this talk, we conclude the measure equivalence classfication and discuss some of the techniques used.
In this talk, I will introduce the extension graph of graph product of groups and explain its geometry. This notion enables us to study the properties of graph product by exploiting the large-scale geometry of its defining graph. In particular, I show that the asymptotic dimension of the extension graph exhibits the same behavior as in the case of quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. In addition, I present applications of the extension graph to the study of convergence actions, graph wreath products, and group von Neumann algebras when the defining graph is hyperbolic.
Given an arbitrary Coxeter system (W,S) and a nonnegative integer m, the m-Shi arrangement of (W, S) is a subarrangement of the Coxeter hyperplane arrangement of (W,S). The classical Shi arrangement (m = 0) was introduced in the case of affine Weyl groups by Shi to study Kazhdan-Lusztig cells for W. As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W and that the union of their inverses form a convex subset of the Coxeter complex. The set of m-low elements in W were introduced to study the word problem of the corresponding Artin-Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in W. In this talk, I will discuss how to Shi’s results extend to any Coxeter system and show that the minimal elements in each Shi region are in fact the m-low elements. This talk is based on joint work with Matthew Dyer, Susanna Fishel and Alice Mark.
A finitely generated group G is called subgroup separable if every finitely generated subgroup H of G is closed in the profinite topology on G (equivalently, there is a family of finite index subgroups of G intersecting in H). One of the initial motivations for studying residually finite groups and subgroup separable groups was McKinsey-Malcev algorithm solving the word problem in finitely presented residually finite groups. Recently, separability has played a crucial role in low-dimensional topology, namely in the resolutions of the Virtually Haken and Virtually Fibered conjectures.
A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that each their finitely generated subgroup H is a retract of a finite-index subgroup K. It also states that K can be obtained from H by a series of free products with infinite cyclic groups. The first statement of the theorem was generalized by Wilton for limit groups. Haglund-Wise proved it for right-angled Artin groups when H is word quasiconvex. We generalize the second statement of Hall's theorem and prove that such K can be obtained from H by a series of certain HNN-extentions. This implies that if L is a right-angled Artin group, H a word quasiconvex subgroup of L, then there is a finite dimensional representation of L that separates the subgroup H in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate H in L. This implies the same statement for a virtually special group L and, in particular, a fundamental group of a hyperbolic 3-manifold. For limit groups this implies similar polynomial bounds and the resolution of the Hanna Neumann conjecture.
These are joint results with K. Brown and A. Vdovina.
Given a Gromov hyperbolic space equipped with an action of a group by isometries, one can study the orbit equivalence relation of the induced action of the group on the Gromov boundary of the space. Marquis and Sabok proved that the action of hyperbolic groups on their Gromov boundaries turns out to have the property that the orbits can be arranged into lines in a consistent manner, a property known as hyperfiniteness. We show that (infinitely presented) graphical small cancellation groups exhibit a similar phenomenon, inducing hyperfinite orbit equivalence relations on the boundaries of their natural hyperbolic Cayley graphs. This is joint work with Damian Osajda and Koichi Oyakawa.
This is joint work with Yeeka Yau. I will explain why minimal elements of cone type components and Shi components form "Garside shadows" in Coxeter groups. This leads to convenient path systems in their Cayley graphs.
This old conjecture asserts that one cannot kill a group by adding one generator and one additional relation. Progress on open problems is often reflected by repeatedly enhancing and strengthening a specific line of attack. Here, however, the progress has instead been reflected by a genuine diversity of ideas and approaches. I will give a survey of the known results and methods proving special cases of this conjecture.