Subspace stabilisers in hyperbolic lattices

In a joint work with Nikolay Bogachev, Alexander Kolpakov and Leone Slavich we show that immersed totally geodesic \(m\)-dimensional suborbifolds of an \(n\)-dimensional arithmetic hyperbolic orbifold correspond to finite subgroups of the commensurator whenever \(m\) is sufficiently large. In particular, for \(n = 3\) this condition includes all totally geodesic suborbifolds.  We call such totally geodesic subspaces finite centraliser subspaces (or fc-subspaces for short) and use them to formulate an arithmeticity criterion for hyperbolic lattices. We show that a hyperbolic orbifold is arithmetic if and only if it has infinitely many fc-subspaces, while in the non-arithmetic case the number of fc-subspaces is finite and bounded in terms of the volume.  We also provide examples of non-arithmetic \(n\)-orbifolds that contain non-fc subspaces of codimension one. A further analysis reveals other deep connections between fc-subspaces and arithmetic/geometric properties of hyperbolic orbifolds.  In the talk I will begin with an introduction to the topic and then discuss some results and their proofs.