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Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This…

We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only…

If $G$ is a centreless group, then $\tau(G)$ denotes the height of the automorphism tower of $G$. We prove that it is consistent that for every cardinal $\lambda$ and every ordinal $\alpha < \lambda$, there exists a centreless group $G$…

Dyer and Formanek (1976) proved that if N is a free nilpotent group of class two and of finite rank which is not equal to 1, or to 3, then the automorphism group Aut(N) of N is complete. The main result of the present paper states that the…

This thesis has three goals related to the automorphism groups of finite $p$-groups. The primary goal is to provide a complete proof of a theorem showing that, in some asymptotic sense, the automorphism group of almost every finite…

An automorphism $\alpha$ of a group $G$ is said to be central if $\alpha$ commutes with every inner automorphism of $G$. We construct a family of non-special finite $p$-groups having abelian automorphism groups. These groups provide counter…

Let $T$ be a tree and $e$ an edge in $T$. If $C$ is a component of $T\setminus e$ and both $C$ and its complement are infinite we say that $C$ is a half-tree. The main result of this paper is that if $G$ is a closed subgroup of the…

Given a group retraction $r: G \rightarrow H $, we construct a finite topological space $ X_r $ of height 1, together with a topological retraction $\overline{r}: X_r \rightarrow X_r $, such that the group of automorphisms $…

The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms…

Let $K$ be a field and $f:\mathbb{P}^N \to \mathbb{P}^N$ a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group $\text{PGL}_{N+1}$. The group of automorphisms, or…

Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horo\v{s}evski\u{i}'s result that every automorphism $\alpha$ of a finite nilpotent group has a cycle whose length…

If f is a nontrivial automorphism of a thick building Delta of purely infinite type, we prove that there is no bound on the distance that f moves a chamber. This has the following group-theoretic consequence: If G is a group of…

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…

We show that every normal amenable subgroup of the automorphism group of the full shift is contained in its center. This follows from the analysis of this group's Furstenberg topological boundary, through the construction of a minimal and…

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least…

It is shown that a group defined by forbidding all patterns of size s+1 that do not appear in a given self-similar group of tree automorphisms is the topological closure of a self-similar, countable, regular branch group, branching over its…

In this article, we show that the stabilized automorphism group of free exact odometers arising from actions of finitely generated residually finite groups coincides with the topological full group of the odometer acting on itself by right…

A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a…

Let $X$ be a projective, connected and smooth scheme defined over an algebraically closed field $k$. In this paper we prove that a tower of finite torsors (i.e., under the action of finite $k$-group schemes) can be dominated by a single…

We work with a finite relational vocabulary with at least one relation symbol with arity at least 2. Fix any integer $m > 1$. For almost all finite structures (labelled or unlabelled) such that at least $m$ elements are moved by some…