Stone’s representation theorem asserts a duality between Boolean algebras on the one hand and Stone spaces, which are compact, Hausdorff, and totally disconnected, on the other. This duality implies a natural isomorphism between the homeomorphism group of the space and the automorphism group of the algebra. We introduce a complex of cuts on which these groups act, and prove that when the algebra is countable and the space has at least five points, that these groups are the full automorphism group of the complex.

We find the shortest realized stretch factor for a fully irreducible $\varphi \in \operatorname{Out}(F_3)$ and show that it is realized by a “principal” fully irreducible element. We also show that it is the only principal fully irreducible produced by a single fold in any rank.

The main result of this paper is that the outer automorphism group of a free product of finite groups and cyclic groups is semistable at infinity (provided it is one ended) or semistable at each end. In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. We also prove that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of finite and cyclic groups is one ended. As a corollary, the outer automorphism groups of the free product of four finite groups or the free product of a single finite group with a free group of rank two are virtual duality groups of dimension two, in contrast with the above example. Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krstić and Vogtmann.

Generalizing Culler and Vogtmann’s Outer Space for the free group, Guirardel and Levitt construct an Outer Space for a free product of groups. We completely characterize when this space (or really its simplicial spine) supports an equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric. Our results are mostly negative, extending thesis work of Bridson and related to thesis work of Cunningham. In particular, provided the dimension of the spine is at least three, it is never CAT(0). Surprisingly, we exhibit one family of free products for which the Outer Space is two-dimensional and does support an equivariant CAT(0) metric.

One version of Whitehead’s famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex–cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.

Dowdall and Taylor observed that given a finite-index subgroup of a free group, taking covers induces an embedding from the Outer Space of the free group to the Outer Space of the subgroup, that this embedding is an isometry with respect to the (asymmetric) Lipschitz metric, and that the embedding sends folding paths to folding paths. The purpose of this note is to extend this result to virtually free groups. We further extend a result of Francaviglia and Martino, proving the existence of “candidates” for the Lipschitz distance between points in the Outer Space of the virtually free group. Additionally we identify a deformation retraction of the spine of the Outer Space for the virtually free group with the space considered by Krstić and Vogtmann.

The fundamental group of a finite graph of groups with trivial edge groups is a free product. We are interested in those outer automorphisms of such a free product that permute the conjugacy classes of the vertex groups. We show that in particular cases of interest, such as where vertex groups are themselves finite free products of finite and cyclic groups, given such an outer automorphism, after passing to a positive power, the outer automorphism is represented by a particularly nice kind of relative train track map called a CT. CTs were first introducd by Feighn and Handel for outer automorphisms of free groups. We develop the theory of attracting laminations for and principal automorphisms of free products. We prove that outer automorphisms of free products satisfy an index inequality reminiscent of a result of Gaboriau, Jaeger, Levitt and Lustig and sharpening a result of Martino. Finally, we prove a result reminiscent of a theorem of Culler on the fixed subgroup of an automorphism of a free product whose outer class has finite order.

In this paper we develop the theory of train track maps on graphs of groups. Expanding a definition of Bass, we define a notion of a map of a graph of groups, and of a homotopy equivalence. We prove that under one of two technical hypotheses, any homotopy equivalence of a graph of groups may be represented by a relative train track map. The first applies in particular to graphs of groups with finite edge groups, while the second applies in particular to certain generalized Baumslag–Solitar groups.

We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to includ extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that automorphism groups of torsion-free one-ended toral relatively hyperbolic groups and certain right-angled Artin groups satisfy our quantitative subgroup alternative.

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

We show the existence of several new infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten’s example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.

Given a finite subgroup $G$ of the mapping class group of a surface $S$, the Nielsen realization problem asks whether $G$ can be realized as a finite group of homomorphisms of $S$. In 1983, Kerckhoff showed that for $S$ a finite-type surface, any finite subgroup $G$ may be realized as a group of isometries of some hyperbolic metric on $S$. We extend Kerckhoff’s result to orientable, infinite-type surfaces. As applications, we classify torsion elements in the mapping class group of the plane minus a Cantor set, and also show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of $S$. Finally, we show that compact subgoups of the mapping class group of $S$ are finite, and locally compact subgroups are discrete.