Automorphic orbits in free groups

Let $F_n$ be the free group of a finite rank $n$. We study orbits $Orb_{\phi}(u)$, where $u$ is an element of the group $F_n$, under the action of an automorphism $\phi$. If an orbit like that is finite, we determine precisely what its cardinality can be if $u$ runs through the whole group $F_n$, and $\phi$ runs through the whole group $Aut(F_n)$. Another problem that we address here is related to Whitehead's algorithm that determines whether or not a given element of a free group of finite rank is an automorphic image of another given element. It is known that the first part of this algorithm (reducing a given free word to a free word of minimum possible length by elementary Whitehead automorphisms) is fast (of quadratic time with respect to the length of the word). On the other hand, the second part of the algorithm (applied to two words of the same minimum length) was always considered very slow. We give here an improved algorithm for the second part, and we believe this algorithm always terminates in polynomial time with respect to the length of the words. We prove that this is indeed the case if the free group has rank 2.Comment: 10 page

Algebraic extensions in free groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.Comment: 35 page

Isometric endomorphisms of free groups

An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries. Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absolute minimizer for relative Gromov norm) for its boundary; this implies, for instance, that a random element of a free group with commutator length at most n has commutator length exactly n and stable commutator length exactly n-1/2. Our methods also let us construct explicit (and computable) quasimorphisms which certify these facts.Comment: 26 pages, 6 figures; minor typographical edits for final published versio

Dual automorphisms of free groups

For any choice of a basis $\cal A$ the free group $F_N$ of finite rank $N \geq 2$ can be canonically identified with the set $F(\cal A)$ of reduced words in $\cal A\cup \cal A^{-1}$. However, such a word $w \in F(\cal A)$ admits a second interpretation, namely as cylinder $C^1_w \subset \partial F_N$. The subset of $\partial F_N$ defined by $C^1_w$ depends not only on the element of $F_N$ given by the word $w$, but also on the chosen basis $\cal A$. In particular one has in general, for \Phi \in \Aut(F_N): $$\Phi(C^1_w) \neq C^1_{\Phi(w)}$$ Indeed, the image of a cylinder under an automorphism \Phi \in \Aut(F_N) is in general not a cylinder, but a finite union of cylinders: $$\Phi (C^1_w)=C^{1}_U := \bigcup_{u_i \in U} C^1_{u_i}$$ In his thesis the first author has given an efficient algorithm and a formula how to determine such a (uniquely determined) finite {\em reduced} set $U = U(w) \subset F_N$. We use those to define the dual automorphism $\Phi_{\cal A}^*$ by setting $\Phi_{\cal A}^*(w) = U(w)$. \smallskip \noindent {\bf Theorem:} {\it For any \Phi \in \Aut(F_N) there are at most 2N distinct finite subsets $U_i \subset F_N$ such that for any $w = y_1 ... y_r \in F_A$ there is one of them, say $U_{i(w)}$, with $$\Phi_{\cal A}^*(w) = \Phi(w) U_{i(w)}\, ,$$ and $U_{i(w)}$ depends only on the last letter y_r \in \CA \cup \CA^{-1}. Furthermore, the seize of each $U_{i}$ is bounded by $2^t$, where $t \geq 0$ is the number of Nielsen automorphisms in any decomposition of $\Phi$ as product of basis permutations, basis inversions and elementary Nielsen automorphisms.

Subset currents on free groups

We introduce and study the space of \emph{subset currents} on the free group $F_N$. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in $F_N$, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis $A$ of $F_N$, a subset current may also be viewed as an $F_N$-invariant measure on a "branching" analog of the geodesic flow space for $F_N$, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of $F_N$ with respect to $A$.Comment: updated version; to appear in Geometriae Dedicat