Activities This Week

A group is said to have a fixed-point property with respect to some class of metric spaces if any isometric action of the group on any space in the class admits a fixed point.

In this talk, I will focus on fixed-point properties with respect to (classes of) Banach spaces. I will survey some results regarding groups with and without these fixed-point properties and then present a recent result of mine regarding fix-point properties for random groups with respect to l^p spaces.

Jacobians of CM curves are abelian varieties with a particularly large endomorphism algebra, which provides them with a rich arithmetic structure. The motivating question for the results in this talk is whether we can find hyperelliptic and non-hyperelliptic curves with maximal CM by a given order whose Jacobians are isogenous. Joint work with Sorina Ionica, and Jeroen Sijsling considers this question in genus 3 by using the catalogue of CM fields in the LMFDB, and found a (small) list of such isogenous Jacobians. This talk describes the main constructions, some results, and Shimura class groups.

Each complete CAT(0) space has an associated topological space, called visual boundary, that coincides with the Gromov boundary in case that the space is hyperbolic. A CAT(0) group G is called boundary rigid if the visual boundaries of all CAT(0) spaces admitting a geometric action by G are homeomorphic. If G is hyperbolic, G is boundary rigid. If G is not hyperbolic, G is not always boundary rigid. The first such example was found by Croke-Kleiner.

In this talk we will see that every group acting freely and cocompactly on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the boundary homeomorphic to a join of two copies of the Cantor set. The proof of this result uses a generalization of classical dynamics on boundaries introduced by Guralnik and Swenson. I will explain the idea of this generalization by explaining a higher-dimensional version of classical North-south-dynamics obtained this way.

This is a joint work with Kasia Jankiewicz, Kim Ruane and Bakul Sathaye.

The radius of comparison of a C-algebra is one measure of the generalization to C-algebras of the dimension of a compact space. Part of the Toms-Winter conjecture says, informally, that a simple separable nuclear unital C*-algebra satisfying the UCT is classifiable if and only if its radius of comparison is zero. Nonzero radius of comparison played a key role in one of the main families of counterexamples to the original form of the Elliott classification program.

It has been known for some time that the radius of comparison of C (X) is, ignoring additive constants, at most half the covering dimension of X. (The factor 1/2 appears because of the use of complex scalars in C*-algebras.) In 2013, Elliott and Niu used Chern character arguments to show that the radius of comparison of C (X) is, again ignoring additive constants, at least half the rational cohomological dimension of X. This left open the question of which dimension the radius of comparison is really related to. The rational cohomological dimension can be strictly less than the integer cohomological dimension, and there are spaces with integer cohomological dimension 3 but infinite covering dimension.

We show that, up to a slightly worse additive constant, the radius of comparison of C (X) is at least half the covering dimension of X. The proof is fairly short and uses little machinery.