Fall 2024

Abstract: We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex, codimension-1 subgroups whose limit sets are locally flat subspheres. The proof is based on ideas used by Markovic in his work on Cannon’s conjecture for cubulated hyperbolic groups with 2-sphere boundary. This is joint work with M. Incerti-Medici.

Abstract: Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism of mapping class groups.  An important theorem of Birman-Hilden and MacLachlan-Harvey says that this lifting map is injective for most finite regular branched covers of surfaces.  Margalit-Winarski asked whether one could prove an analogous result for 3-manifolds.  In this talk, we answer this question by showing that in contrast to the surface case, the lifting map is not injective for most branched covers of 3-manifolds.   This includes double covers of S^3 branched over an unlink, which generalize the hyperelliptic branched covers of S^2; for these covers, we describe the kernel of the lifting map.  In these examples, the lifting map is closely related to a natural map between symmetric automorphism groups of free products.

Abstract: The fundamental group of an n-dimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space H^n. If n is at most 3 or the manifold is arithmetic of simplest type, then the group also admits many geometric actions on CAT(0) cube complexes. I will talk about a joint work with Nic Brody in which we approximate the asymptotic geometry of the action on H^n by actions on these complexes, solving a conjecture of Futer and Wise. The main tool is a codimension-1 generalization of the space of geodesic currents introduced by Bonahon.

Abstract: Relatively hyperbolic groups may admit infinitely many peripheral structures and captures different scales of negative curvature.  Classical work of Stallings and Dunwoody demonstrates that groups containing finite codimension-1 subgroups act on trees where elliptic subgroups form a particular peripheral structure of the group.  Relative cubulations introduces by Einstein and Groves generalize such actions to more general hyperbolic CAT(0) cube complexes and provide new avenues to cubulate groups and prove residual finiteness. In joint work with Einstein and Krishna MS,  we produce peripheral structures for groups containing sufficiently many full codimension-1 subgroups that are elliptic on any dual cube complex. These ideas let us extend the boundary criterion for relative cubulations to allow peripheral subgroups that are not one-ended.

Possible topics: hyperbolic and nonpositively curved groups, boundaries of groups, residual finiteness, etc.

Abstract: Burger-Mozes constructed examples of simple groups acting geometrically on a CAT(0) complex, which is a product of trees. As a counterpoint, we prove that every group acting geometrically (and minimally) on a CAT(0) cube complex which is not a product, admits a nontrivial quotient which also admits a geometric action on a CAT(0) cube complex. Our construction relies on the cubical version of small cancelation theory. This is joint work with M. Arenas and D. Wise.

Abstract: I will survey recent work on the homology of algebras, including joint work with Hepworth and Patzt. In this work, we abstract the notion of homological stability to sequences of algebras, and show that sequences of Temperley-Lieb, Brauer and Partition diagram algebras satisfy homological stability.

Abstract: Motivated by the question of whether braid groups are CAT(0), we investigate CAT(0) behavior of fundamental groups of plane curve complements and certain universal families. If the monodromy of the complement of a plane curve C is finite, we show that CP^2— C admits a complete nonpositively curved metric, and when C is the branch locus of a smooth, complete intersection, we show that the fundamental group of CP^2— C is CAT(0). In the another direction, we prove that the fundamental group of the universal family associated with the singularities of type E_6, E_7, and E_8 is not CAT(0). This is joint work with C. Bregman and A. Libgober.

Abstract: A group is strongly shortcut if it has a Cayley graph in which circles cannot embed at arbitrarily large scales with arbitrarily good bilipschitz constants. This can be shown to be a special case of the Gromov mesh condition implying simply connected asymptotic cones and polynomial Dehn function. Most classes of nonpositively curved groups are strongly shortcut, including CAT(0) groups, Helly groups, systolic groups and hierarchically hyperbolic groups. I will discuss various results on strongly shortcut groups, including recent joint work with Timothy Riley in which we showed that the strong shortcut property is not invariant under change of generating sets.

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