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 n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} 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 Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

Given a Chevalley group G of classical type and a Borel subgroup B ⊆ G, we compute the Σ-invariants of the S-arithmetic groups B(Z[1/N ]), where N is a product of large enough primes. To this end, we let B(Z[1/N ]) act on a Euclidean building X that is given by the product of Bruhat-Tits buildings Xp associated to 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 CAT(0)-spaces to be continuous. We apply these conditions to associate to each simplex at infinity τ ⊂ ∂∞X its so-called parabolic building X τ , 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 ∆ contains an apartment, provided ∆ is thick enough and Aut(∆) acts chamber transitively on ∆.

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.

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.

We prove that the minimal representation dimension of a direct product G of non-abelian groups $$G_1,\ldots ,G_n$$ G 1 , … , G n is bounded below by $$n+1$$ n + 1 and thereby answer a question of Abért. If each $$G_i$$ G i is moreover non-solvable, then this lower bound can be improved to be 2n. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups, we show that branch groups cannot be boundedly generated.