# Members & Guests

## Prof. Dr. Stefan Witzel

Professor
JLU Gießen

E-mail: stefan.witzel(at)math.uni-giessen.de
Homepage: https://switzel.eu

Working areas

## Project

8Parabolics and invariants

## Publications within SPP2026

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

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 $$A_{\Gamma}$$.

 Journal Journal of group theory Link to preprint version Link to published version

Related project(s):
8Parabolics and invariants

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