Related papers: A classification theorem for boundary 2-transitive…

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same…

An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G…

Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying…

Let $(X, \sigma)$ be a transitive sofic shift and let $\rm{Aut}(X)$ denote its automorphism group. We generalize a result of Frisch, Schlank, and Tamuz to show that any normal amenable subgroup of $\rm{Aut}(X)$ must be contained in the…

In this note, we show that locally finite quasi-transitive graphs are quasi-isometric to trees if and only if every other locally finite quasi-transitive graph quasi-isometric to them is minor excluded. This generalizes results by…

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite…

Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching…

Let (S, B) be the log pair associated with a projective completion of a smooth quasi-projective surface V . Under the assumption that the boundary B is irreducible, we obtain an algorithm to factorize any automorphism of V into a sequence…

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras $\O_n$ which preserve both the diagonal and the core $UHF$-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is…

We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant…

This article proves a version of the Feit-Thompson theorem for simple groups of finite Morley rank: a connected groups of finite Morley rank with a finite Sylow 2-subgroup has a trivial Sylow 2-subgroups.

A finite tree $T$ with $|V(T)| \geq 2$ is called {\it automorphism-free} if there is no non-trivial automorphism of $T$. Let $\mathcal{AFT}$ be the poset with the element set of all finite automorphism-free trees (up to graph isomorphism)…

There are several natural families of groups acting on rooted trees for which every member is known to be amenable. It is, however, unclear what the elementary amenable members of these families look like. Towards clarifying this situation,…

In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an…

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and…

A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all…

It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected…

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to…