Date:
Thu, 26/01/202310:00-11:00
Title: From geometry to arithmetic of hyperbolic orbifolds
Abstract: The problem of detecting arithmeticity of a hyperbolic n-manifold only through its geometry has been remained open in general for several decades. This question attracted many famous experts (including Reid and McMullen), but only very recently several revolutionary solutions were suggested to this problem. Different superrigidity approaches were suggested by Margulis and Mohammadi in dimension 3, and by Bader, Fisher, Miller and Stover for all n>2.
In a joint paper with Misha Belolipetsky, Sasha Kolpakov and Leone Slavich we also developed a large industry connecting geometry and arithmetic of hyperbolic orbifolds and manifolds. We introduce a new class of the so-called fc-subspaces and use them to formulate an arithmeticity criterion for hyperbolic orbifolds: a hyperbolic orbifold M is arithmetic if and only if it has infinitely many fc-subspaces. We also show that immersed totally geodesic m-dimensional suborbifolds of n-dimensional arithmetic hyperbolic orbifolds are fc-subspaces whenever m \ge \floor{n/2}, and we provide examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. One of the key results of our paper is a characterization of immersions of arithmetic hyperbolic orbifolds into one another.
In the talk I will give an overview of this area and present our main results. I will also provide a few examples (from reflection groups and knot theory) as a motivation for this new class of fc-subspaces. If time permits, we will briefly discuss our methods.
Abstract: The problem of detecting arithmeticity of a hyperbolic n-manifold only through its geometry has been remained open in general for several decades. This question attracted many famous experts (including Reid and McMullen), but only very recently several revolutionary solutions were suggested to this problem. Different superrigidity approaches were suggested by Margulis and Mohammadi in dimension 3, and by Bader, Fisher, Miller and Stover for all n>2.
In a joint paper with Misha Belolipetsky, Sasha Kolpakov and Leone Slavich we also developed a large industry connecting geometry and arithmetic of hyperbolic orbifolds and manifolds. We introduce a new class of the so-called fc-subspaces and use them to formulate an arithmeticity criterion for hyperbolic orbifolds: a hyperbolic orbifold M is arithmetic if and only if it has infinitely many fc-subspaces. We also show that immersed totally geodesic m-dimensional suborbifolds of n-dimensional arithmetic hyperbolic orbifolds are fc-subspaces whenever m \ge \floor{n/2}, and we provide examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. One of the key results of our paper is a characterization of immersions of arithmetic hyperbolic orbifolds into one another.
In the talk I will give an overview of this area and present our main results. I will also provide a few examples (from reflection groups and knot theory) as a motivation for this new class of fc-subspaces. If time permits, we will briefly discuss our methods.