Explicit canonical cycle at the virtual cohomological dimension of $\mathrm{SL}_n(\mathbb{Z})$ through Voronoi complex

Abstract

We construct an explicit canonical cycle in the top-dimensional homology of the Voronoi complex associated with an arithmetic group. This cycle relates to the cohomology of SL$_n(\mathbb{Z})$ with rational coefficients at the virtual cohomological dimension. This cycle has been previously identified in computational works and conjectured to provide an intrinsic generator. Our approach relies on a geometric rigidity property of Voronoi tessellations. Furthermore, an abstract framework for polyhedral tessellations of convex cones under group actions is established, elucidating the underlying mechanism of the construction of such cycles.