We show that if a finitely generated group $G$ has a nonelementary WPD action on a hyperbolic metric space $X$ , then the number of $G$ -conjugacy classes of $X$ -loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$ . As an application we prove that for $N\geq 3$ the number of distinct $\text{Out}(F_{N})$ -conjugacy classes of fully irreducible elements $\unicode[STIX]{x1D719}$ from an $R$ -ball in the Cayley graph of $\text{Out}(F_{N})$ with $\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$ of the order of $R$ grows exponentially in $R$ .

Motivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index function fprim(n, FN) to the residual finiteness growth function for FN.

We prove that for $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb P\mathit{Curr}(F_k)$ that is either $\mathit{Out}(F_k)$-equivariant or $\mathit{Out}(F_k)$-anti-equivariant. Here $\partial CV(F_k)$ is the ‘length function’ boundary of Culler–Vogtmann's Outer space $CV(F_k)$, and $\mathbb P\mathit{Curr}(F_k)$ is the space of projectivized geodesic currents for $F_{k}$. We also prove that, if $k\ge 3$, for the action of $\mathit{Out}(F_k)$ on $\mathbb P\mathit{Curr}(F_{k})$ and for the diagonal action of $\mathit{Out}(F_k)$ on the product space $\partial CV(F_k)\times \mathbb P\mathit{Curr}(F_k)$, there exist unique non-empty minimal closed $\mathit{Out}(F_k)$-invariant sets. Our results imply that for $k\ge 3$ any continuous $\mathit{Out}(F_k)$-equivariant embedding of $CV(F_k)$ into $\mathbb P\mathit{Curr}(F_k)$ (such as the Patterson–Sullivan embedding) produces a new compactification of Outer space, different from the usual ‘length function” compactification $\overline{CV(F_k)}=CV(F_k)\cup \partial CV(F_k)$.

We obtain a number of results regarding the freeness of subgroups of Coxeter groups, Artin groups and one-relator groups with torsion. In the case of Coxeter groups, we also obtain results on quasiconvexity and subgroup separability.

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G*, such that G is a subgroup of G* but G is not quasiconvex in G*. We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank 2 which is malnormal and quasiconvex in G.

We show that if A is a torsion-free word hyperbolic group which belongs to class (Q), that is all finitely generated subgroups of A are quasiconvex in A, then any maximal cyclic subgroup U of A is a Burns subgroup of A. This, in particular, implies that if B is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then A *UB, ⧼A, t | Ut = V⧽ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting 3-manifold groups are known to belong to class (Q) and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.

Analogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.