Quasi-trees and product set growth

In this thesis we investigate the geometric properties of quasi-trees, and study product set growth in groups with acylindrical actions on quasi-trees and hyperbolic spaces. The main tool we use when considering quasi-trees is the end-approximating tree, which is a tree that we can construct from a geodesic space while preserving its space of ends. We use this construction to prove that every quasi-tree is (1,C)-quasi-isometric to a simplicial tree, one consequence of which is that having a cobounded quasi-action on a simplicial tree is equivalent to having a quasi-conjugate (1,C)-quasi-action on a simplicial tree. The end-approximating tree can also be used to show that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives us a natural extension of the standard construction of a visual metric on the boundary of a tree. As another consequence, we are able to show that Gromov's Tree Approximation Lemma for hyperbolic spaces can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. This uniform version of Gromov's Tree Approximation Lemma has an application to Delzant and Steenbock's work on product set growth in acylindrically hyperbolic groups. In particular, we are able to show that for any group G that acts acylindrically on a quasi-tree, there exists α>0 such that for any finite U ⊂ G that has large enough displacement in the quasi-tree, and is not contained in a virtually cyclic subgroup, we have that |Un|⩾(α|U|)⸤n+1/2⸥. By a result of Balasubramanya, this theorem applies to every acylindrically hyperbolic group. We also study growth in acylindrically hyperbolic groups directly from their actions on hyperbolic spaces, which allows us to get a dichotomy on the finitely generated subgroups of mapping class groups, at the cost of working only with symmetric subsets. Specifically, we show that for every surface S of finite type, there exist α, β > 0 such that for any finite symmetric subset U of the mapping class group MCG(S) we have that |Un|⩾(α|U|)βn, so long as no finite index pure subgroup of h 〈U〉 has an infinite order central element. As right-angled Artin groups embed as subgroups of mapping class groups, a similar dichotomy also applies in this case. We separately prove that we can quickly generate loxodromic elements in right-angled Artin groups, which by a result of Fujiwara shows that the set of growth rates for many of their subgroups are well-ordered.