Gilbert Levitt, Martin Lustig
We show that every automorphism $α$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $α^q$ such that every element of the compactum $F_k \cup \partial F_k$ converges to a fixed point under iteration of $α^q$.
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Periodic automorphisms of free groups are diagonalisable in free skew-fields
本文研究自由斜域上的弱周期扭乘自同构，证明由自由群的周期自同构诱导的自由斜域自同构必等价于对角自同构。
Regular automorphisms of periodic $p'$-groups need not be splitting
该研究针对周期p'-群，证明了正则自同构与分裂自同构的概念并不一致，通过构造特定周期群给出了否定答案。
In this paper we describe the automorphism groups of the endomorphism semigroups of free Burnside groups $B(m,n)$ for odd exponents $n\ge1003$. We prove, that the groups $Aut(End(B(m,n)))$ and $Aut(B(m,n))$ are isomorphic. In particular, if the groups $Aut(End(B(m,n)))$ and $Aut(End(B(k,n)))$ are isomorphic, then $m=k$.
We study properties of eigenvalues of a matrix associated with a randomly chosen partial automorphism of a regular rooted tree. We show that asymptotically, as the numbers of levels goes to infinity, the fraction of non-zero eigenvalues converges to zero in probability.
Periodic points and residual finiteness of automorphism groups of subshifts
研究子移位的周期点密度与其自同构群余有限性之间的关系，证明在周期点不密集时，自同构群非余有限，并构造了一个非余有限的强不可约子移位例子。
A constant spinal group is a subgroup of the automorphism group of a regular rooted tree, generated by a group of rooted automorphisms $A$ and a group of directed automorphisms $B$ whose action on a subtree is equal to the global action. We provide two conditions in terms of certain dynamical systems determined by $A$ and $B$ for constant spinal groups to be periodic, generalising previous results on Grigorchuk--Gupta--Sidki groups and other related constructions. This allows us to provide various new examples of finitely generated infinite periodic groups.
Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups
该研究探讨周期阿贝尔群的自同态环与自同构群的初等等价问题，给出了自同态环初等等价的充要条件，并针对特定周期阿贝尔群刻画了自同构群初等等价的等价条件。
We consider the outer automorphism group Out(A_Gamma) of the right-angled Artin group A_Gamma of a random graph Gamma on n vertices in the Erdos--Renyi model. We show that the functions (log(n)+log(log(n)))/n and 1-(log(n)+log(log(n)))/n bound the range of edge probability functions for which Out(A_Gamma) is finite: if the probability of an edge in Gamma is strictly between these functions as n grows, then asymptotically Out(A_Gamma) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out(A_Gamma) is almost surely infinite. T…
We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.
We prove that finitely generated virtually free groups are stable in permutations. As an application, we show that almost-periodic almost-automorphisms of labelled graphs are close to periodic automorphisms.
We prove that an automorphism $ϕ:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.
We define fully irreducible automorphisms of generalized Baumslag-Solitar groups in analogy with fully irreducible automorphisms of free groups. We first obtain a characterization of fully irreducible automorphisms analogous to a condition given by Kapovich. Next we discuss the existence of pseudo-periodic conjugacy classes based on the study of Nielsen paths in a train track representative. Finally we give an algorithm which finds out whether an automorphism with a train track representative is pseudo-atoroidal and fully irreducible.
The goal of this paper is to introduce a new tool, called {\em long turns}, which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.
We study the group $IAut(A)$ generated by the inertial automorphisms of an abelian group $A$, that is, automorphisms $γ$ with the property that each subgroup $H$ of $A$ has finite index in the subgroup generated by $H$ and $Hγ$. Clearly, $IAut(A)$ contains the group $FAut(A)$ of finitary automorphisms of $A$, which is known to be locally finite. In a previous paper, we showed that $IAut(A)$ is (locally finite)-by-abelian. In this paper, we show that $IAut(A)$ is also metabelian-by-(locally finite). In particular, $IAut(A)$ has a normal subgroup $Γ$ such that $IAut(A)/Γ$ is locally finite and $…
Subperiodic groups and bounded automorphisms of periodic graphs
研究亚周期群在三维空间中的分类，通过开发易计算的不变量识别同构类，并探讨有限阶有界自同构在晶格图中的应用。
We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional condition, they are periodic).
We describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.
We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism groups which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree. Our results further cover several interesting finitely generated subgroups of the tree almost automorphism groups, including the Thompson groups $F$, $T$, and $V$. We show in particular that Thompson's group $T$ has no commensurated subgroups other…
We study the loxodromic elements for the action of $Out(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. We prove that each outer automorphism is either loxodromic, or has bounded orbits without any periodic point, or has a periodic point; and we prove that all three possibilities can occur. We also prove that two loxodromic elements are either co-axial or independent, meaning that their attracting/repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each of the alternatives in these results is also characte…
We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of a birational action of an infinite Coxeter group on ordered point sets in the projective plane modulo projective equivalence. We also outline the classification of non-rational surfaces with large automorphism groups.
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