Adam Piggott, Kim Ruane, Genevieve S. Walsh
We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
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The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on.
We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.
We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_\circ^\circ$-subgroup.
Let $ϕ$ be an automorphism of a free group $F_n$ of rank $n$, and let $M_ϕ=F_n \rtimes_ϕ \mathbb{Z}$ be the corresponding mapping torus of $ϕ$. We study the group $Out(M_ϕ)$ under certain technical conditions on $ϕ$. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of $ϕ$ in $GL_2(\mathbb{Z})$.
Symmetric polytopes whose automorphism groups are 2-groups
研究了具有2-群自同构群的多面体，证明了其自同构群的继承性及扩展性质。
We construct an example of an IA-automorphism of the free metabelian group of rank $n\geq 3$ without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain. By a result of Bachmuth \cite{Ba1}, such an automorphism does not exist if the rank is equal to 2.
Automorphisms of a Free Centre-by-Centre-by-Metabelian Group of Rank 3
研究3阶自由中心-中心-可解群的自同构群，发现其包含一个稠密于形式幂级数拓扑的有限生成子群。
The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct w…
It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear automorphisms. We prove that it is not true for an arbitrary torsion subgroup. We construct an example of abelian $p$-group of automorphism of polynomial algebra of rank 2 over the field of complex numbers, which is not conjugated with a subgroup of linear automorphisms.
In this paper we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm{IA}_3$ of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of $\mathrm{IA}_3$, which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the low…
We improve a result of Prokhorov and Shramov on the rank of finite $p$-subgroups of the birational automorphism group of a rationally connected variety. Known examples show that they are sharp in many cases.
Automorphisms of Groups and a Higher Rank JSJ Decomposition II: The single ended case
本文研究满足弱无柱条件的层级双曲群(HHG)的自同构，通过构造高秩JSJ分解来编码单个自同构的动力学及HHG外自同构群的结构。
Let $Γ_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(Γ_d(q))$ and $Out(Γ_d(q))$ for $d \geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d \geq 3$. Specifically, we show that when $d \geq 3$, the groups $Γ_d(q)$ have property $R_{\infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $Γ_2(q)=L_q = {\mathbb Z}_q \wr {\mathbb Z}$ h…
Automorphism groups of solvable groups of finite abelian ranks
本文研究了可解群的自同构群结构，通过有理代数包络的构造，探讨了S-算术子群的性质及外自同构群的算术像。
We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an $ω$-categorical structure can go down to reducts. Together, those results prove that a large number of $ω$-categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any $ω$-categorical structure to which they do not apply (assuming the automorphism group has no compact quotie…
In this paper we define a quantity called the rank of an outer automorphism of a free group which is the same as the index introduced in [D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, `An index for counting fixed points for automorphisms of free groups', Duke Math. J. 93 (1998) 425-452] without the count of fixed points on the boundary. We proceed to analyze outer automorphisms of maximal rank and obtain results analogous to those in [D.J. Collins and E. Turner, `An automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element', J. Pure and App…
A survey article that presents some recent algebraic and model-theoretic results on the automorphism groups of relatively free groups of infinite rank. The topics include topological aspects, generating sets, descripition of automorpisms and expressive power of the first-order theories.
The automorphism groups of small affine rank 3 graphs
本文研究仿射秩3图的自同构群，通过确定有限多个较小图的完整自同构群并结合已知结果，将一般问题归结为基座为非交换单群的情形。
Let $F_n$ be a free group of rank $n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$. A word $u$ of $F_n$ is called positive if $u$ does not have negative exponents. A word $u$ in $F_n$ is called potentially positive if $ϕ(u)$ is positive for some automorphism $ϕ$ of $F_n$. We prove that there is an algorithm to decide whether or not a given word in $F_2$ is potentially positive, which gives an affirmative solution to problem F34a in [1] for the case of $F_2$. Two elements $u$ and $v$ in $F_n$ are said to be boundedly translation equivalent if the ratio …
On the outer automorphism groups of free groups
该研究证明了秩可数无穷的自由群的外自同构群具有完备性，丰富了自由群外自同构群相关的代数理论结论。
The Automorphism Group of the Infinite-Rank Free Group is Coarsely Bounded
本文证明了可数无限秩自由群的完全自同构群和外部自同构群是粗有界的，即它们在度量空间上不存在具有无界轨道的连续作用，且具有一个点的拟等距类型。
We describe, up to degree equal to the rank, the Lie algebra associated with the automorphism group of a free group. We compute in particular the ranks of its homogeneous components, and their structure as modules over the linear group. Along the way, we infirm (but confirm a weaker form of) a conjecture by Andreadakis, and answer a question by Bryant-Gupta-Levin-Mochizuki.
In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been observed on many occasions that the most anomalous examples invariably have zero p-rank. In this paper, the K-automorphism group Aut(X) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that if the curve has genus g greater than or equal to 2, and |Aut(X)|>24g^2, then Aut(X) ha…
We introduce a new class of automorphisms $φ$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov homeomorphism of a surface with arbitrary many boundary components. More generally, there may be subgroups of $F_N$ of rank $\geq 2$ on which $φ$ restricts to the identity. We prove some basic facts about such {\em tree-irreducible} automorphisms, and show that, together with Dehn twist automorphisms, they are the natural basic building blocks from which any automorphism of …
A fully irreducible outer automorphism phi of the free group F_n of rank n has an expansion factor which often differs from the expansion factor of the inverse of phi. Nevertheless, we prove that the ratio between the logarithms of the expansion factors of phi and its inverse is bounded above by a constant depending only on the rank n. We also prove a more general theorem applying to an arbitrary outer automorphism of F_n and its inverse, and their entire spectrum of expansion factors.
Flat groups of automorphisms of totally disconnected, locally compact groups
研究完全不连通、局部紧致群的平坦自同构群，揭示其结构特征及高秩群的可能来源。
Small groups of finite Morley rank with a supertight automorphism
本文研究具有有限莫利秩和 Prüfer 2-秩1的无限单群，在超紧自同构作用下固定点子群均为伪有限的条件下，证明该群同构于某特征不为2的代数闭域上的 PGL₂(K)。
In universal algebraic geometry the category of the finite generated free algebras of some fixed variety of algebras and the quotient group A/Y are very important. Here A is a group of all automorphisms of this category and Y is a group of all inner automorphisms of this category. In the varieties of all the groups, all the abelian groups (see B. Plotkin and G. Zhitomirski, 2006), all the nilpotent groups of the class no more then n (see A. Tsurkov, 2007) the group A/Y is trivial. B. Plotkin posed a question: "Is there a subvariety of the variety of all the groups, such that the group A/Y in t…
The knot quandles of Suciu's ribbon $n$-knots and automorphisms on the free group of rank two
本文通过分析秩二自由群自同构的共轭类，为 Suciu 的带状 n-纽结的纽结拟阵互不同构提供了新证明。
Taking the High-Edge Route Through Outer Space in Rank 3
本文研究了主完全不可约外自同构的train track maps，证明了哪些秩3图承载这些映射，并得出哪些秩3 Culler-Vogtmann Outer space单纯形被主轴穿过的推论。
Let $G$ be a group. The orbits of the natural action of Aut$(G)$ on $G$ are called ``automorphism orbits'' of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. We prove that if $G$ is a soluble group with finite rank such that $ω(G)< \infty$, then $G$ contains a torsion-free characteristic nilpotent subgroup $K$ such that $G = K \rtimes H$, where $H$ is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that $ω(G)=3$.
For each prime power $q\geq 5$, we construct a rank four chiral polytope that has a group $PSL(3,q)$ as automorphism group and Schläfli type $\{q-1,\frac{2(q-1)}{(3,q-1)},q-1\}$. We also construct rank five polytopes for some values of $q$ and we show that there is no chiral polytope of rank at least six having a group $PSL(3,q)$ or $PSU(3,q)$ as automorphism group.
If H is a flat group of automorphisms of finite rank n of a totally disconnected, locally compact group G, then each orbit of H in the metric space B(G) of compact, open subgroups of G is quasi-isometric to n-dimensional euclidean space. In this note we prove the following partial converse: Assume that G is a totally disconnected, locally compact group such that B(G) is a proper metric space and let H be a group of automorphisms of G such that some (equivalently every) orbit of H in B(G) is quasi-isometric to n-dimensional euclidean space, then H has a finite index subgroup which is flat of ra…
In this work we construct free Moufang loop in the variety generated by code loops. We apply this construction for study the code loops. Moreover, we define and determine all basic representations of code loops of rank 3 and 4.
Automorphism groups of cocompact CAT(0) cube complexes and simplicity
本文研究特殊余紧 CAT(0) 立方体复形的自同构群，给出其生成集与大小二分法，并在额外条件下证明 Aut^+ 为单、非离散 tdlc 子群。
We prove that, for any positive integer $c$, the quotient group $γ_{c}(M_{3})/γ_{c+1}(M_{3})$ of the lower central series of the McCool group $M_{3}$ is isomorphic to two copies of the quotient group $γ_{c}(F_{3})/γ_{c+1}(F_{3})$ of the lower central series of a free group $F_{3}$ of rank $3$ as $\mathbb{Z}$-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra ${\rm gr}(M_{3})$ of $M_3$ is naturally embedded into the Johnson Lie algebra ${\cal L}({\rm IA}(F_{3}))$ of the IA-automorphisms of $F_{3}$.
This paper uses the methods from arXiv:0804.1396 to give 'short' presentations for Aut$^+(F_n)$, the special automorphism group of the free group of rank $n$, and Out $^+(F_n)$ the special outer automorphism group of the free group of rank $n$.
The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. Babai conjectured that if a primitive coherent configuration with $n$ vertices is not a Cameron scheme, then its automorphism group has minimal degree $\geq cn$ for some constant $c>0$. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3. In this pap…
We exhibit normal subgroups of a free nilpotent group F of rank two and class three, which have isomorphic finite quotients but are not conjugate under any automorphism of F.
We prove that for any automorphism $α$ of a free group F of finite rank, one can efficiently compute a basis of the fixed point subgroup Fix(α).
It is already known that the automorphism group of a chiral polyhedron is never isomorphic to $PSL(2,q)$ or $PGL(2,q)$ for any prime power $q$. In this paper, we show that $PSL(2,q)$ and $PGL(2,q)$ are never automorphism groups of chiral polytopes of rank at least $5$. Moreover, we show that $PGL(2,q)$ is the automorphism group of at least one chiral polytope of rank $4$ for every $q\geq5$. Finally, we determine for which values of $q$ the group $PSL(2,q)$ is the automorphism group of a chiral polytope of rank $4$, except when $q=p^d\equiv3\pmod{4}$ where $d>1$ is not a prime power, in which c…
Automorphismes du groupe des automorphismes d'un groupe de Coxeter universel
本文利用自由积的Guirardel-Levitt外空间，确定了秩$n \geq 4$的泛Coxeter群的自同构群及外自同构群的外自同构群结构。
A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on poi…
Proper partial linear spaces affording imprimitive rank 3 automorphism groups
本研究针对真部分线性空间的对称分类问题，首次完成了半本原非本原秩3真部分线性空间的分类，涵盖本原传递、拟本原群情形，构造了多族无限实例及10个单独实例，多数为文献中新结果。
It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category $\mathsf{Group}^{\mathcal{J}}$ of p…
The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities: alg-rk(W)\le flat-rk(G)\le rk(|W|\_0). Here, alg-rk(W) is the maximal $\mathbb{Z}$-rank of abelian subgroups of W, and rk(|W|\_0) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|\_0. We can prove these inequalities under weaker assumptions. We also sh…
We prove that every simplicial automorphism of the free splitting graph of a free group of at least rank 3 is induced by an outer automorphism of the free group.
We prove that, for n=3 and 4, the minimal nonabelian finite factor group of the outer automorphism group Out F_n of a free group of rank n is the linear group PSL_n(Z_2) (conjecturally, this may remain true for arbitrary rank n > 2). We also discuss some computational results on low index subgroups of Aut F_n and Out F_n, for n = 3 and 4, using presentations of these groups.
Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes
研究CAT(0)立方复形的自同构群与Hagen接触图的自同构群的等价性，揭示接触图与Kim-Koberda扩展图的差异。
We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.
It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if $A$ is finite. In particular, this applies to dessins d'enfants, regarded as finite oriented hypermaps. The proof, involving maximal subgroups of various triangle groups, yields a simple construction of a regular map whose automorphism group contains an isomorphic copy of every finite group.
In this article, we give a method of calculating the automorphism groups of the vertex operator algebras $V_L^+$ associated with even lattices $L$. For example, by using this method we determine the automorphism groups of $V_L^+$ for even lattices of rank one, two and three, and even unimodular lattices.
We study groups of bimeromorphic and biholomorphic automorphisms of projective hyperkähler manifolds. Using an action of these groups on some non-positively curved space, we deduce many of their properties, including finite presentation, strong form of Tits' alternative and some structural results about groups consisting of transformations with infinite order.
In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form $n$ over a field $k$ of characteristic not two, and a category arising from an action of the projective similarity group of $n$ on certain pairs of automorphisms of the group scheme $\mathbf{PGO}^+(n)$ defined over $k$. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.
Automorphisms of free metabelian Lie algebras, I
本文研究了自由双交换李代数的自同构性质，证明了特定条件下自同构的 tame 性及部分自同构的几乎 tame 性，推翻了领域内长期存在的结论。
Considering a particular case of a problem posed by Saharon Shelah, we prove that the automorphism group of an infinitely generated free nilpotent group N first-order interprets the full second-order theory of the set rank(N) in the empty language.
We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of Handel-Mosher axis bundles that are not just a single axis, as well as of "nongeneric" behavior in the sense of the "train track directed" random walk of Kapovich-Pfaff. The tools developed here also allow one to construct a whole new array of ideal Whitehead graphs achieved by ageometric fully irreducible outer automorphisms in all ranks.
We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}Γ\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.
In this paper we prove that every automorphism of (elementary) adjoint Chevalley group with root system of rank $>1$ over a commutative ring (with 1/2 for the systems $A_2$, $F_4$, $B_l$, $C_l$; with 1/2 and 1/3 for the system $G_2$) is standard, i.\,e., it is a composition of ring, inner, central and graph automorphisms.
Consider unbounded reduced Abelian p-groups (p > 2) A and A'. In this paper, we prove that if the automorphism groups Aut A and Aut A' are elementary equivalent then the groups A and A' are equivalent in the second order logic bounded by the final rank of the basic subgroups of A and A'.
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