A tighter bound for the number of words of minimum length in an automorphic orbit

Fibers of automorphic word maps and an application to composition factors

In this paper, we study the fibers of "automorphic word maps", a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite groups $G$ where, for some fixed nonempty reduced word $w$ in $d$ variables and some fixed $ρ\in\left(0,1\right]$, the word map $w_G$ on $G$ has a fiber of size at least $ρ|G|^d$: No sufficiently large alternating group and no (classical) simple group of Lie type of sufficiently high rank can occur as a composition factor of such a group $G$.

自守词映射与Amit--Ashurst猜想

Automorphic word maps and Amit--Ashurst conjecture

本文通过构造与给定词同分布的自守词映射，改进并部分解决了有限幂零群上的Amit--Ashurst猜想，同时推广了Camina等人及Iniguez和Sangroniz的相关结果。

Counting words of minimum length in an automorphic orbit

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…

Automorphic orbits in free groups: words versus subgroups

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…

Automorphic orbits in free groups

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 …

Classification of automorphic conjugacy classes in the free group on two generators

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.