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Given a finitely generated group $G$ that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ is conjugate into a peripheral subgroup. As an application, we obtain a trichotomy for subgroups of relatively hyperbolic groups. Moreover we prove the existence of the relative exponential growth rate for all subgroups of limit groups.

Related project(s):
8Parabolics and invariants

Given a Chevalley group \(\mathcal{G}\) of classical type and a Borel subgroup \(\mathcal{B} \subseteq \mathcal{G}\), we compute the \(\Sigma\)-invariants of the \(S\)-arithmetic groups \(\mathcal{B}(\mathbb{Z}[1/N])\), where \(N\) is a product of large enough primes. To this end, we let \(\mathcal{B}(\mathbb{Z}[1/N])\) act on a Euclidean building \(X\) that is given by the product of Bruhat-Tits buildings \(X_p\) associated to \(\mathcal{G}\), where \(p\) runs over the primes dividing \(N\). In the course of the proof we introduce necessary and sufficient conditions for convex functions on \(\mbox{CAT}(0)\)-spaces to be continuous. We apply these conditions to associate to each simplex at infinity \(\tau \subset \partial_{\infty} X\) its so-called parabolic building \(X^{\tau}\), which we study from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential \(n\)-connectivity rather than actual \(n\)-connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building \(\Delta\) contains an apartment, provided \(\Delta\) is thick enough and \(\mbox{Aut}(\Delta)\) acts chamber transitively on \(\Delta\).

Related project(s):
8Parabolics and invariants

The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to $\Z$ whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type $\F_2$ but not $\FP_3$, and examples where the RACG is hyperbolic and the kernel is finitely generated and non-hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz--Norin--Wise involving Bestvina--Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.

Related project(s):
8Parabolics and invariants

We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate \(\lim \limits_{n \rightarrow \infty} \sqrt[n]{\vert {B^{X}_H(n)} \vert}\) of a subgroup \(H\) of a finitely generated acylindrically hyperbolic group $G$ exists with respect to every finite generating set \(X\) of \(G\), if \(H\) contains a loxodromic element of \(G\). Further we prove that the relative exponential growth rate of every finitely generated subgroup \(H\) of a right-angled Artin group \(A_{\Gamma}\) exists with respect to every finite generating set of \(A_{\Gamma}\).

Related project(s):
8Parabolics and invariants