An Improper Surface Group Action
In 1999, Long and Reid proposed a proper action of a surface group on a product of trees. In this note, we show that the action is not proper.
math.GR↗
arXiv subjects
Nic Brody
Publications and source records attributed to Nic Brody.
Approximating hyperbolic lattices by cubulations
We show that an isometric action of a torsion-free uniform lattice $\Gamma$ on hyperbolic space $\mathbb{H}^n$ can be metrically approximated by geometric actions of $\Gamma$ on $\mathrm{CAT}(0)$ cube complexes, provided that either $n$ is at most three, or the lattice is arithmetic of simplest type. This solves a conjecture of Futer and Wise. Our main tool is the study of a space of co-geodesic currents, consisting of invariant Radon measures supported on codimension-1 hyperspheres in the Gromov boundary of $\mathbb{H}^n$. By pairing co-geodesic currents and geodesic currents via an intersection number, we show that asymptotic convergence of geometric actions can be deduced from the convergence of their dual co-geodesic currents. For surface groups, our methods also imply approximation by cubulations for actions induced by non-positively curved Riemannian surfaces with singularities, Hitchin and maximal representations, and quasiFuchsian representations.
math.GR↗
Generalized Residual Finiteness of Groups
A countable group is residually finite if every nontrivial element can act nontrivially on a finite set. When a group fails to be residually finite, we might want to measure how drastically it fails - it could be that only finitely many conjugacy classes of elements fail to act nontrivially on a finite set, or it could be that the group has no nontrivial actions on finite sets whatsoever. We define a hierarchy of properties, and construct groups which become arbitrarily complicated in this sense.
math.GR↗
Greenberg-Shalom's Commensurator Hypothesis and Applications
We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the $`70$s and more generally by Shalom in the early $2000$s. We refer to this positive answer as the Greenberg-Shalom hypothesis. This hypothesis then says that any infinite discrete subgroup of a semisimple Lie group with dense commensurator is a lattice in a product of some factors. For some applications it is natural to extend the hypothesis to cover semisimple algebraic groups over other fields as well.
math.GR↗
Building Groups From Restricted Diagrams of Groups
We consider the problem of realizing a group as the fundamental group of a graph of groups where the vertex groups are restricted to certain classes (for example, coming from a certain finite list of groups, or having bounded geometric rank). We show how this places restrictions on the possible groups that can be realized and we give a topological application of our results to the problem of constructing manifolds from a finite set of "building blocks".
math.GT↗