We examine the palindromic automorphism group $ΠA(F_n)$ of a free group $F_n$, a group first defined by Collins which is related to hyperelliptic involutions of mapping class groups, congruence subgroups of $SL_n(\Z)$, and symmetric automorphism groups of free groups. Cohomological properties of the group are explored by looking at a contractible space on which $ΠA(F_n)$ acts properly with finite quotient. Our results answer some conjectures of Collins and provide a few striking results about the cohomology of $ΠA(F_n)$, such as that its rational cohomology is zero at the vcd.
In this paper, we initiate the study of palindromic automorphisms of groups that are free in some variety. More specifically, we define palindromic automorphisms of free nilpotent groups and show that the set of such automorphisms is a group. We find a generating set for the group of palindromic automorphisms of free nilpotent groups of step 2 and 3. In particular, we obtain a generating set for the group of central palindromic automorphisms of these groups. In the end, we determine central palindromic automorphisms of free nilpotent groups of step 3 which satisfy the necessary condition of Br…
Let $F_n$ be the free group of rank $n$ with free basis $X=\{x_1,\dots,x_n \}$. A palindrome is a word in $X^{\pm 1}$ that reads the same backwards as forwards. The palindromic automorphism group $ΠA_n$ of $F_n$ consists of those automorphisms that map each $x_i$ to a palindrome. In this paper, we investigate linear representations of $ΠA_n$, and prove that $ΠA_2$ is linear. We obtain conjugacy classes of involutions in $ΠA_2$, and investigate residual nilpotency of $ΠA_n$ and some of its subgroups. Let $IA_n$ be the group of those automorphisms of $F_n$ that act trivially on the abelianisatio…
We introduce the palindromic automorphism group and the palindromic Torelli group of a right-angled Artin group A_G. The palindromic automorphism group Pi A_G is related to the principal congruence subgroups of GL(n,Z) and to the hyperelliptic mapping class group of an oriented surface, and sits inside the centraliser of a certain hyperelliptic involution in Aut(A_G). We obtain finite generating sets for Pi A_G and for this centraliser, and determine precisely when these two groups coincide. We also find generators for the palindromic Torelli group.
A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group ΠA_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group ΠA_n has close connections with hyperelliptic mapping class groups, braid groups, congruence subgroups of GL(n,Z), and symmetric automorphisms of free groups. We obtain a generating set for the subgroup of ΠA_n consisting of those elements acting trivially on the abelianisation of F_n, the palindromic Torelli gr…
We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalgamation of groups; extreme properties of hyperbolic 3-orbifold groups, relations between inner and quasi-inner automorphisms of groups; and Staic's construction of symmetric cohomology of groups.
We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $G$ in $\HH^3$ whether or not $G$ is discrete. We show that there is a {\sl core geodesic} $Ł$ in the convex hu…
In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.
The present article constitutes the third part of our study of the large scale geometry of metrisable groups, the first two part appearing in the companion paper "Large scale geometry of metrisable groups". In this third part, we present a detailed study of the theory in the special case of non-Archimedean Polish groups, that is, automorphism groups of countable first order structures.
A scale-multiplicative semigroup in a totally disconnected, locally compact group $G$ is one for which the restriction of the scale function on $G$ is multiplicative. The maximal scale-multiplicative semigroups in groups acting 2-transitively on the set of ends of trees without leaves are determined in this paper and shown to correspond to geometric features of the tree.
In the 1970s Stallings showed that one could learn a great deal about free groups and their automorphisms by viewing the free groups as fundamental groups of graphs and modeling their automorphisms as homotopy equivalences of graphs. Further impetus for using graphs to study automorphism groups of free groups came from the introduction of a space of graphs, now known as Outer space, on which the group $Out(F_n)$ acts nicely. The study of Outer space and its $Out(F_n)$ action continues to give new information about the structure of $Out(F_n)$, but has also found surprising connections to many o…
We define a self-map Pal: F_2 --> F_2 of the free group on two generators a, b, using automorphisms of F_2 that form a group isomorphic to the braid group B_3. The map Pal restricts to de Luca's right iterated palindromic closure on the submonoid generated by a, b, and is continuous for the profinite topology on F_2. The values of Pal are palindromes and coincide with the elements g of F_2 such that abg is conjugate to bag.
We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer Space as a model for Out(F_n), we prove that the natural combinatorial and topological model for their outer automorphism groups can \emph{not} be given an equivariant CAT(0) metric. This is particularly interesting as there are very few non-trivial examples of proving that a mod…
This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product of certain elementary automorphisms (inner automorphisms, partial conjugations, automorphisms associated to symmetries of the underlying graph). This allows us to completely compute the automorphism group of certain graph products, for instance in the case where the underlying graph is finite, connected, leafless and of girth at least $5$. If in addition the…
We present a new simple proof of the fact that certain group manifolds as well as certain homogeneous spaces G/H of dimension 4n admit a quaternionic triple of integrable complex structures that are covariantly constant with respect to the same torsionful Bismut connection, i.e. exhibit the HKT geometry. The key observation is that different complex structures are interrelated by automorphisms of the Lie algebra. To construct the quaternion triples, one only needs to construct the proper automorphisms, which is a more simple problem.
This article is a survey on ind-varieties and ind-groups introduced by Shafarevich in 1965, with a special emphasis on automorphism groups of affine varieties and actions of ind-groups on ind-varieties. We give precise definitions and complete proofs, including several known results. The survey contains many examples and also some questions which came up during our work on the subject. Among the new results we show that for an affine variety X the automorphism group Aut(X) is always locally closed in the ind-semigroup End(X) of all endomorphisms, and we give an example of a strict closed subgr…
We investigate the cocompact action of Higman's group on a CAT(0) square complex associated to its standard presentation. We show that this action is in a sense intrinsic, which allows for the use of geometric techniques to study the endomorphisms of the group, and show striking similarities with mapping class groups of hyperbolic surfaces, outer automorphism groups of free groups and linear groups over the integers. We compute explicitly the automorphism group and outer automorphism group of Higman's group, and show that the group is both hopfian and co-hopfian. We actually prove a stronger r…
For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli geometry of our surface and prove that every automorphism of the Torelli geometry is induced by a diffeomorphism of the surface in question. We also provide an intrinsic algebraic characterization of some natural elements of the Torelli group and of some geometric relations between them. When combined, these results allow us to compute the automorphism group…
We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension of this group depending on the transverse geometry are found. Further, we investigate Cartan foliations covered by fibrations. When the global holonomy group of that foliation is discrete, we obtain the explicit new formula for determining its basic automorphism Lie group. Example…
For a fully irreducible automorphism ϕof the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'ϕ^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'ϕ^n for n large.
We consider transformations preserving certain linear structure in Grassmannians and give some generalization of the Fundamental Theorem of Projective Geometry and the Chow Theorem [Ch]. It will be exploited to study linear $(k,n-k)$-involutions, $1<k<n-1$. The analogy of the well-known J. Dieudonné and C. E. Rickart result will be obtained.
We investigate automorphism groups of planar graphs. The main result is a complete recursive description of all abstract groups that can be realized as automorphism groups of planar graphs. The characterization is formulated in terms of inhomogeneous wreath products. In the proof, we combine techniques from combinatorics, group theory, and geometry. This significantly improves the Babai's description (1975).
Scissors automorphism groups II: Solomon-Tits theorems
University of Toronto Scarborough（多伦多大学士嘉堡分校）；University of Aberdeen（阿伯丁大学）；Binghamton University（宾汉姆顿大学）；Purdue University（普渡大学）；Universität Münster（明斯特大学）
We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex C on which the tame automorphism group acts naturally. We prove that C is contractible and Gromov-hyperbolic, and we prove that Tame($k^3$) is acylindrically hyperbolic by finding explicit loxodromic weakly proper discontinuous elements.
Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F_4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Moufang quadrangle of type F_4 stabilizing the absolute points of a polarity, also centralizes the polarity. This handles the final case of a similar result for all polarities of Moufang n-gons.
This is the expanded version of my talk at the workshop "Groups of Automorphisms in Birational and Affine Geometry", October 29--November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. The appendix is the expanded version of my notes on open problems posted on the site of this workshop. It contains formulations of some open problems and the relevant comments.
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 object of this expository work is to try to unveil the topological/geometric intuition behind the theory of free groups and their automorphism and outer automorphism groups. The method we follow is to focus on a series of problems in the study of free groups, and use the solutions of those problems to motivate topological/geometric tools. We do not aim to write down proofs which minimize the number of alphanumeric characters. We instead strive to write down proofs which maximize the development of broadly applicable geometric tools. In Part I we study problems solved by Nielsen and Whitehe…
We give a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group $L_n$ has infinitely many twisted conjugacy classes for any automorphism $\vp$ only when $n$ is divisible by 2 or 3, originally proved by Gonçalves and Wong. We determine when the wreath product $G \wr \Z$ has this same property for several classes of finite groups $G$, including symmetric groups and some nilpotent groups.
This is a survey of the Kawamata-Morrison cone conjecture on the structure of Calabi-Yau varieties and more generally Calabi-Yau pairs. We discuss the proof of the cone conjecture for algebraic surfaces, with plenty of examples. We show that the automorphism group of a K3 surface need not be commensurable with an arithmetic group, which answers a question by Mazur.
This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [Pl],[Ts]) we must consider the variety, to which our algebras belongs, the category K of all finitely generated free algebras of our variety and research how the group of all the automorphisms of this category AutK are different from the group of the all inner automorphisms of this category InnK. An automorphism of the category is called inner, if it is isomorphic as functor to the ide…
