Suppose a residually finite group $G$ acts cocompactly on a contractible complex with strict fundamental domain $Q$, where the stabilizers are either trivial or have normal $mathbb{Z}$-subgroups. Let $partial Q$ be the subcomplex of $Q$ with nontrivi
al stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of $G$. We show that independent of the chain, the normalized torsion limits to the torsion of $partial Q$, shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual $L^2$-torsion of $G$ in terms of the torsion of stabilizers and topology of $partial Q$. In particular, we get complete answers for right-angled Artin groups, which shows they satisfy a torsion analogue of the Luck approximation theorem.
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible s
pherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of locally reducible Artin groups, which includes all $2$-dimensional Artin groups, and for spherical Artin groups of any type other than $E_6$, $E_7$, $E_8$. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.
Bestvina introduced a $mathcal{Z}$-structure for a group $G$ to generalize the boundary of a CAT(0) or hyperbolic group. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $mathc
al{E}mathcal{Z}$-structure. In this paper, we show that fundamental groups of graphs of nonpositively curved Riemannian $n$-manifolds admit $mathcal{Z}$-structures and graphs of negatively curved or flat $n$-manifolds admit $mathcal{E}mathcal{Z}$-structures. This generalizes a recent result of the first two authors with Tirel, which put $mathcal{E}mathcal{Z}$-structures on Baumslag-Solitar groups and $mathcal{Z}$-structures on generalized Baumslag-Solitar groups.
The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds $M$ with $b_1(M) = 2$, and whose fundamental groups admit presentat
ions with two generators and one relator. We show that even among this special class, there are 3-manifolds such that the unit ball of the Thurston norm has arbitrarily many faces.
