Covering theory for complexes of groups

We develop an explicit covering theory for complexes of groups, parallel to that developed for graphs of groups by Bass. Given a covering of developable complexes of groups, we construct the induced monomorphism of fundamental groups and isometry of universal covers. We characterize faithful complexes of groups and prove a conjugacy theorem for groups acting freely on polyhedral complexes. We also define an equivalence relation on coverings of complexes of groups, which allows us to construct a bijection between such equivalence classes, and subgroups or overgroups of a fixed lattice $\Gamma$ in the automorphism group of a locally finite polyhedral complex $X$.Comment: 47 pages, 1 figure. Comprises Sections 1-4 of previous submission. New introduction. To appear in J. Pure Appl. Algebr

Asymptotic distribution of values of isotropic quadratic forms at SS-integral points

We prove an analogue of a theorem of Eskin-Margulis-Mozes: suppose we are given a finite set of places $S$ over $\mathbb{Q}$ containing the archimedean place and excluding the prime $2$, an irrational isotropic form ${{\mathbf q}}$ of rank $n\geq 4$ on $\mathbb{Q}_S$, a product of $p$-adic intervals $I_p$, and a product $\Omega$ of star-shaped sets. We show that unless $n=4$ and ${{\mathbf q}}$ is split in at least one place, the number of $S$-integral vectors ${\mathbf v} \in {\mathsf{T}} \Omega$ satisfying simultaneously ${\mathbf q}( {\mathbf v} ) \in I_p$ for $p \in S$ is asymptotically given by $$\lambda({\mathbf q}, \Omega) | I| \cdot \prod_{p\in S_f} T_p^{n-2},$$ as ${\mathsf{T}}$ goes to infinity, where $| I |$ is the product of Haar measures of the $p$-adic intervals $I_p$. The proof uses dynamics of unipotent flows on $S$-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an $S$-arithmetic variant of the $\alpha$-function introduced in the work of Eskin, Margulis, Mozes, and an $S$-arithemtic version of a theorem of Dani-Margulis.Comment: The article will appear in Journal of Modern Dynamics. In the revised version, several typos have been fixed. Moreover, another preprint (arXiv:1605.02436) has been incorporate