Related papers: Homomorphisms from automorphism groups of free gro…

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses…

We define two complexes on which the group Aut$(F_n)$ acts freely. The homotopy groups of these are studied. They map to the K-groups of Z and are themselves a sort of pre-K-theory.

We study the cluster automorphism group $Aut(\mathcal{A})$ of a coefficient free cluster algebra $\mathcal{A}$ of finite type. A cluster automorphism of $\mathcal{A}$ is a permutation of the cluster variable set $\mathscr{X}$ that is…

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…

Let F_n be the free group of rank n and let Aut^+(F_n) be its special automorphism group. For an epimorphism pi : F_n -> G of the free group F_n onto a finite group G we call Gamma^+(G,pi) = {f in Aut^+(F_n) | pi*f = pi} the standard…

Let T be a d-regular tree (d > 2) and A=Aut(T), its automorphism group. Let G be a group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of G has finitely many fixed points on T.

We compute the groups $H^*(\mathrm{Aut}(F_n); M)$ and $H^*(\mathrm{Out}(F_n); M)$ in a stable range, where $M$ is obtained by applying a Schur functor to $H_\mathbb{Q}$ or $H^*_\mathbb{Q}$, respectively the first rational homology and…

Let F_n be the free group on n generators. Define IA_n to be group of automorphisms of F_n that act trivially on first homology. The Johnson homomorphism in this setting is a map from IA_n to its abelianization. The first goal of this paper…

Let $F_n$ be the free group on $n\ge 2$ elements and $\A(F_n)$ its group of automorphisms. In this paper we present a rich collection of linear representations of $\A(F_n)$ arising through the action of finite index subgroups of it on…

We investigate the size of fixed point sets of automorphisms of bounded domains in $\mathbb{C}^n$. In one complex variable, a nontrivial automorphism has at most two fixed points, but in higher dimensions fixed point sets need not be…

We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety.…

A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. The groups $\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which $\operatorname{Aut}(A)$ are…

We calculate the homology of automorphism groups of free groups in various cases. The author would like to thank his thesis advisor Karen Vogtmann.

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…

A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group ${\rm Aut}(F_n)$ of the free group $F_n$ of rank $n$. The automorphism groups of such varieties are…

Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications,…

Let $F_n$ be the free group on $n \geq 2$ generators. We show that for all $1 \leq m \leq 2n-3$ (respectively for all $1 \leq m \leq 2n-4$) there exists a subgroup of $\operatorname{Aut}(F_n)$ (respectively $\operatorname{Out}(F_n)$) which…

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively…

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C_3 which is maximal…

Let $A=A(x_{1},...,x_{n})$ be a free associative algebra in $\mathcal{A}$ freely generated over $K$ by a set $X=\{x_{1},...,x_{n}\}$, $End A$ be the semigroup of endomorphisms of $A$, and $Aut End A$ be the group of automorphisms of the…