Spring 2023

Abstract: Whenever one attempts to make precise the notion of "fundamental group at infinity", one runs into a certain technical issue involving basepoints.  It turns out that the role of basepoint is played by a proper ray, i.e., a proper map from [0,∞) into our space.  But change of basepoint only makes sense between two rays that are joined by a proper homotopy.  Mihalik and Geoghegan have popularized the notion of a space that is "semistable at infinity", which means that all proper rays are properly homotopic.  A finitely presented group is semistable if the universal cover of a finite presentation complex is semistable.

Semistability turns out to be quite mysterious.  An incredibly large number of families of interesting groups are known to be semistable, and there is not a single known example of a non-semistable group!  Geoghegan and Mihalik have asked whether every finitely presented group is semistable.  In this talk, I will give an introductory survey of semistability from the point of view of geometric group theory.

Abstract: Divergence of a metric space is an interesting quasi-isometry invariant of the space which measures how geodesic rays diverge outside of a ball of radius r, as a function of r. Divergence of a finitely generated group is defined as the divergence of its Cayley graph. For symmetric spaces of non-compact type the divergence is either linear or exponential, and Gromov suggested that the same dichotomy should hold in a much larger class of non-positively curved CAT(0) spaces. However this turned out not to be the case and we now know that the spectrum of possible divergence functions on groups is very rich. In a joint project with Pallavi Dani, Yusra Naqvi, and Anne Thomas, we initiate the study of the divergence in the general Coxeter groups. We introduce a combinatorial invariant called the `hypergraph index', which is computable from the Coxeter graph of the group, and use it to characterize when a Coxeter group has linear, quadratic or exponential divergence, and also when its divergence is bounded above by a polynomial.

Abstract: Coarse structures are used to study the large scale geometry of a space.  For example, although the integers and the real line are different topologically, they look the same from "far away", in the sense that any geometric configuration in the real line can be approximated by one in the integers, up to some uniformly bounded error.  A coarse group is a group object in the category of coarse spaces, for example, this means the group operation is only "coarsely associative," etc.  In joint work with Federico Vigolo we study coarse groups.  This talk will be an advertisement for our work, as we walk through examples that illustrate some of our main results, and connections to other subjects. 

Abstract: The connected sum is a standard operation for creating a new $n$-manifold $M\#N$ from of a pair of existing $n$-manifolds $M$ and $N$. Well-definedness of this operation is slightly delicate and requires some care. When working with noncompact manifolds, an alternative operation called the end-sum is often preferable. Well-definedness (or lack thereof) of end-sum is a topic of current interest. In this talk I will introduce some lesser-known algebraic machinery---the end (co)homology---that is useful for studying noncompact spaces in general and end-sums in particular. The main goal of the talk will be a description of some new results illustrating the delicate nature of detecting differences between end-sums of manifolds.

Abstract: Covering spaces and surfaces are ubiquitous in topology, but there are still some fundamental questions whose answers are not widely known.  In particular, one can ask; given two surfaces, when does one admit a finite-sheeted cover over the other.

If the surfaces are closed, this reduces to an exercise in Euler characteristics.  A result of Massey provides the answer when the surfaces are finite-type with boundary.  In joint work with Ty Ghaswala, we look at what can be said about the remaining (uncountably many) cases.

Abstract: In 1997 Delzant observed that fundamental groups of hyperbolic manifolds with large injectivity radius have nicely behaved group rings. In particular, these rings have no zero divisors and only the trivial units. In this talk I will discuss an extension of this observation showing that such rings have a division algorithm (generalizing the division algorithm for group rings of free groups discovered by Cohn) and ``freedom theorems’’ saying ideals generated by two elements are free (which can be viewed as generalizations from subgroups to ideals of some freedom theorems of Delzant and Gromov). This has geometric consequences to the homotopy classification of 2-complexes with surface fundamental groups and to complexity of cell structures on hyperbolic manifolds.

Abstract: We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a CAT(0) cube complex. As a consequence, we prove that if $\Gamma$ is a non-uniform lattice in a non-compact semisimple Lie group $G$ that admits a relatively geometric action on a CAT(0) cube complex, then $G$ is isomorphic to $\SO(n,1)$. We also prove that given a relatively hyperbolic group with residually finite parabolic subgroups, if it is K\"ahler and acts relatively geometrically on a CAT(0) cube complex, then it is virtually a surface group. This work is joint with Corey Bregman and Daniel Groves.

Abstract: Cutting a hyperbolic surface along a simple closed multi-geodesic yields a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of hyperbolic surface. This result strengthens Mirzakhani’s famous simple close multi-geodesic counting theorems for hyperbolic surfaces. This is joint work with Aaron Calderon.

Abstract: Hyperbolic Coxeter groups with Sierpinski carpet boundary was investigated by Swiatkowski. And hyperbolic right-angled Coxeter group with Gromov boundary as Menger ´ curve was studied by Daniel Danielski. Also, Haulmark, Hruska, and Sathaye’s produced the first known examples of non-hyperbolic CAT(0) groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on n vertices for n greater than or equal to 5. Recently, Danielski and Swiatkowski gave complete characterizations (in terms of nerves) of the word hyperbolic ´ Coxeter groups whose Gromov boundary is homeomorphic to the Sierpi´nski curve and to the Menger curve, respectively. In our presentation, we find new examples with both hyperbolic 4 and nonhyperbolic groups which state: Genus one surface with one boundary component admits a flag triangulation $L$ such that $W_L$ has a Menger curve boundary, and $W_L$ can be non-hyperboic. The construction in Haulmark, Hruska, and Sathaye’s paper depended on a slight extension of Sierpinski’s theorem on embedding 1–dimensional planar compacta into the Sierpinski carpet. However, our methods depend on a perturbing trick for paths and special techniques for nullity condition; also, we exploit good properties of Pontryagin surface. 

Abstract: Among asymptotic properties of groups, the action of groups on their boundaries are a particularly significant class of invariants. The question of what kind of boundary-like actions a group admits, and whether one action can be deformed into another is particularly rich. For instance, the study of quasi-conformal deformations of certain hyperbolic groups on their Gromov boundaries gives rise both to Teichmuller theory and Mostow's rigidity theorem. 

There is a different theory of boundaries of groups that predates Mostow's celebrated theorem. In the mid 1960's Furstenberg introduced the notion of a Stationary Boundary of a random walk on a group, and used this to prove that a cocompact lattice subgroup in a rank 1 simple Lie group cannot be isomorphic to a cocompact lattice in a rank n simple Lie group for any n>2. Since then, the theory of stationary boundary actions of lattices in simple higher rank Lie groups has been completely understood via Margulis:  There are finitely many stationary boundaries, so they admit no non-trivial deformations. 

The problem remains to try to understand the boundaries of rank 1 cocompact lattices, or more generally, hyperbolic groups. We establish that every hyperbolic group has a continuum of boundary actions, and joint with Alex Furman, we see that for certain isometry groups of the hyperbolic plane, stationary boundaries admit high dimensional continuous families of deformations. 

Abstract: What does a random quotient of a hyperbolic group look like? Ollivier studied the density model of quotients of a hyperbolic group. The density parameter d measures the proportion of the Cayley ball picked as relators. Further, he provided a proof  that for d <1/2, a random quotient of a non-elementary hyperbolic group in this model remains non-elementary hyperbolic.

In this talk, we will introduce a new ( closely related ) density model of random quotients called the frayed-geodesic model.  We will also show that in this model, for d< 1/2  a random quotient of a hyperbolic group also remains non-elementary hyperbolic.

Abstract: When does an ascending chain of free subgroups of a given group G stabilize? Without placing conditions on the rank of each entry in the chain not much can be said. The story is quite different for ascending chains of bounded rank. G. Higman and M. Takahashi independently proved that when G is a free group every such chain stabilizes. I. Kapovich and A. Miyasnikov re-phrased their proof in the language of Stallings’ folds. This proof can be generalized to graphs-of-groups where the free subgroups of the vertex groups satisfy the chain condition. As a result every ascending chain of bounded rank free subgroups of a surface group stabilizes. In this talk I will prove that every ascending chain of bounded rank free subgroups in a closed (or finite-volume hyperbolic) 3—manifold group. Hyperbolic geometry, geometrization, and the JSJ decomposition all play a role in the proof. This is joint work with N. Lazarovich.

Abstract: The commensurator of a Riemannian manifold M encodes symmetries between all the finite covers of M, and lifts to a subgroup of isometries of the universal cover of M. In case M is an (irreducible) finite volume locally symmetric space, the commensurator is thus a subgroup of a semisimple Lie group G. Margulis proved that if the commensurator is dense in G, then M is arithmetic. Greenberg-Shalom asked if the same is true for infinite volume M? I will report on recent progress on this question when M regularly covers a finite volume hyperbolic manifold. This is joint work with D. Fisher and M. Mj.

Abstract: For a locally-finite CAT(0) cube complex X, we will carefully describe a family of nested finite subcomplexes {C_i}. These subcomplexes will be shown to have the following properties: 1. each C_i is convex; 2. C_i+1 collapses to C_i for all i; 3. {C_i} gives a compact exhaustion of X.

Abstract: The marked length spectrum of a negatively curved metric space can be thought of as a length assignment to every geodesic of the metric space. A celebrated result by Otal says that for negatively curved closed surfaces, the marked length spectrum completely determines the metric. In my talk, I will discuss my work towards extending Otal’s result to a class of surface amalgams, which can arise as quotients of model geometries of right-angled Coxeter groups.

Abstract: We address the problem of when two solenoidal manifolds are homeomorphic. We first give the most basic example which illustrates the ideas. Then show how this problem can be formulated in 4 equivalent ways: 

(1) as a problem of homeomorphism of foliated spaces; 

(2) as conjugacy between transversals for Cantor actions; 

(3) as commensuration between groups with prescribed group chains; 

(4) as equivalence of group actions on pointed trees. 

Each of these different vantage points give rise to invariants which can be applied to the problem. As an illustration, we describe the Steinitz orders associated to solenoidal manifolds, and the type invariants they define.

Abstract: A theorem in Gulbrandsen’s dissertation led us to consider general questions on the use of inverse limits to obtain nice compactifications. That first led to the consideration of inverse sequences of compacta in which the bonding maps are retractions, and then to a special type of retraction which we call a topological collapse. 

I will review classical examples of inverse limit spaces and some related issues regarding topologies on simplicial complexes. I will also recall the classical theory of expansions and collapses and extend those notions to topological versions. Finally, I will tie these topics to the problem of obtaining Z-compactifications of certain spaces, such as CAT(0) cube complexes. Our work has connections to work by Chapman, Siebenmann, Ferry, Marsh, Prajs and others, which will be explored. The talk will be aimed at a general audience of topologists, including graduate students.

Abstract: The Cartan-Hadamard theorem states that the universal covering space of a connected complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to Rn. More specifically, this diffeomorphism turns out to be the exponential map and it is a covering map of the manifold. If this manifold is also simply connected, then the exponential map is a diffeomorphism from Rn to the manifold itself. This exponential map is usually not a coarse map, but its inverse is a coarse map. In this talk I will walk through the proof of how you use the exponential map and its inverse to get a coarse homotopy equivalence between a manifold with these specific properties and Rn. 

Abstract: Consider a hyperbolic group acting on a real tree. On the one hand, we say that it has the ``bounded bactracking property'' (BBP) if its orbit map is a one-sided quasi-isometry.

This notion was introduced for free groups by Gaboriau-Jaeger-Levitt-Lustig in 1998. On the other hand, we say the action is ``small'' if it has (virtually) cyclic edge stabilizers.

In joint work with Misha Kapovich we show that for a large class of hyperbolic groups, small actions have the BBP. Then we relate this to the problem of extending the translation length of the action to geodesic currents, the space of

measures on the square of the Gromov boundary. Our work complements recent results of Cantrell--Oregon-Reyes.

Abstract: For any non-empty compactum X, the space of embeddings Emb(X,R^N) is separable. An immediate consequence is the following known result: Let F be an uncountable family of mutually disjoint homeomorphic copies of a compactum X in R^N; then there exists a sequence from F converging homeomorphically to an element of F.

In 1970, J.W.Cannon and S.G.Wayment raised the question: Suppose that X_0, X_1, X_2,... is a sequence of pairwise disjoint continua in R^N that converges homeomorphically to X_0. Does it exist an uncountable family of pairwise disjoint homeomorphic copies of X_0 in R^N? Cannon and Wayment obtained a positive answer under the additional assumption: the X_i's are embedded in R^N equivalently to each other.

But even under this assumption, it is not always possible to find a desired uncountable family so that all of its elements are embedded in R^N equivalently to X_0. This is confirmed by examples. For N=2, such examples can be found in the papers of J.H.Roberts (1930) and R.H.Bing (1951).

For N=3 or N>4, Cannon and Wayment constructed a sequence X_0, X_1, X_2,... of pairwise disjoint wild (N-1)-spheres in R^N such that: { X_i } converges homeomorphically to X_0; all X_i's are embedded equivalently to each other; but it is impossible to find uncountably many pairwise disjoint (N-1)-spheres in R^N embedded equivalently to X_0. Here, the impossibility is derived from the results of R.H.Bing for N=3 (1957-1961) and of J.L.Bryant for N>4 (1968). In its turn, the result of Bryant was based on the theorem of A.V.Chernavsky (1968). For N>4, the impossibility can also be derived from the homotopy-theoretic criterion for local flatness proved independently by A.V.Chernavsky (1973) and R.Daverman (1973).

The case N=4 was not covered by the paper of Cannon and Wayment, but the same examples work: for the proof, use the 4-dimensional local flatness criterion by F.Quinn (1982); I am indebted to F.Ancel for this reference.

In the talk, assuming N>3, we construct new series of examples: for (N-1)-spheres; for a wider class of compacta of positive dimension; and for Cantor sets. Our proof is simple, it does not use strong results such as local flatness criterion. Instead, we use ''sticky'' Cantor sets in R^N, N>3, described by V.Krushkal (2016) as an answer to a problem of R.J.Daverman: these sets cannot be isotoped off themselves by small ambient isotopies.