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We investigate a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem. We verify the pro-$p$ version of the conjecture, as well as its higher dimensional abstract analogues.
In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography.
Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $tilde X$ with compact quotient $X$. Fix $n geq 1$, assume $H_n(tilde X, mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the
We show that for every subset X of a closed surface M^2 and every basepoint x_0, the natural homomorphism from the fundamental group to the first shape homotopy group, is injective. In particular, if X is a proper compact subset of M^2, then pi_1(X,x
Classifying endotrivial kG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G, has been a long-running quest, which by deep work of Dade, Alperin, Carlson, Thevenaz, and others, has been reduced
The characteristic index of a locally compact connected group $G$ is the non-negative integer $d$ for which we have a homeomorphism $Gcong Ktimes mathbb{R}^d$ with $Kle G$ maximal compact. We prove that the characteristic indices of closed connected