Let $F_n$ be the free group of a finite rank $n$. We study orbits $Orb_ϕ(u)$, where $u$ is an element of the group $F_n$, under the action of an automorphism $ϕ$. 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 $ϕ$ 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.
We present an effective algorithm for detecting automorphic orbits in free groups, as well as a number of algorithmic improvements of train tracks for free group automorphisms.
Let u be a cyclic word in a free group F_n of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: |v|=|u| and v= ϕ(u) for some ϕ\in \text {Aut}F_n}. In this paper, we prove that N(u) is bounded by a polynomial function with respect to |u| under the hypothesis that if two letters x, y occur in u, then the total number of x and x^{-1} occurring in u is not equal to the total number of y and y^{-1} occurring in u. A complete proof without the hypothesis would yield the polynomial time complexity of Whitehead's al…
A loop is automorphic if all its inner mappings are automorphisms. We construct the free commutative automorphic $2$-generated loop of nilpotency class $3$. It has dimension $8$ over the integers.
Let u be a cyclic word in a free group F_n of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: |v|=|u| and v=ϕ(u) for some ϕ\in AutF_n}. In this paper, we prove that N(u) is bounded by a polynomial function of degree 2n-3 with respect to |u| under the hypothesis that if two letters x, y occur in u, then the total number of x and x^{-1} occurring in u is not equal to the total number of y and y^{-1} occurring in u. We also prove that 2n-3 is the sharp bound on the degree of polynomials bounding N(u). As a sp…
We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F_2? Moreover, decidability subsists if we allow H to be a rational subset of F_2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, we show the decidability of the following weaker problem: given a finitely generated subgroup H, a word u and an integer k, d…
We say a subset $Σ\subseteq F_N$ of the free group of rank $N$ is \emph{spectrally rigid} if whenever $T_1, T_2 \in \cv_N$ are $\mathbb{R}$-trees in (unprojectivized) outer space for which $|σ|_{T_1} = |σ|_{T_2}$ for every $σ\in Σ$, then $T_1 = T_2$ in $\cv_N$. The general theory of (non-abelian) actions of groups on $\mathbb{R}$-trees establishes that $T \in \cv_N$ is uniquely determined by its translation length function $|\cdot|_T \colon F_N \to \mathbb{R}$, and consequently that $F_N$ itself is spectrally rigid. Results of Smillie and Vogtmann \cite{MR1182503}, and of Cohen, Lustig, and St…
We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL(n), Sp(4), and G_2, we perform an analysis of the expansions on split real forms of E_6 and E_7 where simplifications take place for automorphic realizations of real representations which have small Gelfand-Kirillov dimension. Though the character varieties are more complicated for exceptional groups, we explain how the nonvanishing Fourier coefficients for small representations behave analogously to Fou…
We associate a finite directed graph with each equivalence class of words in $F_2$ under $\operatorname*{Aut} F_2$, and we completely classify these graphs, giving a structural classification of the automorphic conjugacy classes of $F_2$. This classification refines work of Khan and proves a conjecture of Myasnikov and Shpilrain on the number of minimal words in an automorphic conjugacy class whose minimal words have length $n$, which in turn implies a sharp upper bound on the running time of Whitehead's algorithm for determining whether two words in $F_2$ are automorphic conjugates.
It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is red…
We prove that there is an algorithm to determine whether a tuple of elements in a toral relatively hyperbolic group G is in the automorphic orbit of the other tuple.
We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts ``like an automorphism" on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological properties of the corresponding one-relator group. These elements (``generalized primitive elements"), interesting in their own right, can also be used for distinguishing automorphisms among arbitrary endomo…
It is well-known that a point $T\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichmüller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup …
We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a {\em representative element} in a {\em repeat-free subgroup}, and for finite abelian groups we can count the number of automorphism classes of elements.
The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem -- to decide whether a word is an element of some basis of the free group -- and the free factor problem can …
This is a survey of recent progress in several areas of combinatorial algebra. We consider combinatorial problems about free groups, polynomial algebras, free associative and Lie algebras. Our main idea is to study automorphisms and, more generally, homomorphisms of various algebraic systems by means of their action on ``very small" sets of elements, as opposed to a traditional approach of studying their action on subsystems (like subgroups, normal subgroups; subalgebras, ideals, etc.) We will show that there is a lot that can be said about a homomorphism, given its action on just a single ele…
Let $G$ be a group. An element $g \in G$ is called a test element of $G$ if for every endomorphism $φ:G \to G$, $φ(g)=g$ implies that $φ$ is an automorphism. We prove that for a finitely generated profinite group $G$, $g \in G$ is a test element of $G$ if and only if it is not contained in a proper retract of $G$. Using this result we prove that an endomorphism of a free pro-$p$ group of finite rank which preserves an automorphic orbit of a non-trivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-$p$ groups and Demushkin groups. By relating t…
In this paper we discuss a genetic version (GWA) of the Whitehead's algorithm, which is one of the basic algorithms in combinatorial group theory. It turns out that GWA is surprisingly fast and outperforms the standard Whitehead's algorithm in free groups of rank >= 5. Experimenting with GWA we collected an interesting numerical data that clarifies the time-complexity of the Whitehead's Problem in general. These experiments led us to several mathematical conjectures. If confirmed they will shed light on hidden mechanisms of Whitehead Method and geometry of automorphic orbits in free groups.
It is shown that every accessible group which is integrable orbit equivalent to a free group is virtually free. Moreover, we also show that any integrable orbit-equivalence between finitely generated groups extends to their end compactifications.
For $N\geq 4$, we show that there exist automorphisms of the free group $F_N$ which have a parabolic orbit in $\partial F_N$. In fact, we exhibit a technology for producing infinitely many such examples.
We prove that if $Γ$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line.
We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ({\FI}) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first $L^2$-Betti number. We obtain Bass-Serre rigidity results, \textit{i.e.} forms of uniqueness in free product decompositions of equivalence relations with ({\FI}) components. The main features of our work are weak algebraic assumptions a…
