Related papers: Big free groups acting on $\Lambda$-trees

Let $\Lambda$ be an ordered abelian group, $\mathrm{Aut}^+(\Lambda)$ the group of order-preserving automorphisms of $\Lambda$, $G$ a group and $\alpha:G\to\mathrm{Aut}^+(\Lambda)$ a homomorphism. An $\alpha$-affine action of $G$ on a…

We will show that all inverse limits of finite rank free groups index by the natural numbers are isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups…

A $f\colon\mathbb{R}\to\mathbb{R}$ is called Hamel function if its graph is a Hamel basis of the linear space $\mathbb{R}^2$ over rationals. We construct, assuming CH, a free group of the size $2^\mathfrak{c}$ contained in the class of all…

We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its…

We describe homomorphisms $\varphi:H\rightarrow G$ for which the codomain is acylindrically hyperbolic and the domain is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in…

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly…

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0)…

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…

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…

The paper is devoted to study the structure of Hawaiian groups of some topological spaces. We present some behaviors of Hawaiian groups with respect to product spaces, weak join spaces, cone spaces, covering spaces and locally trivial…

We consider $\omega^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $\omega^n$ for some integer $n\geq 1$. We show that all these structures are…

In this note we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Graham Higman, in his…

A topological group $G$ is said to have a local $\omega^\omega$-base if the neighbourhood system at identity admits a monotone cofinal map from the directed set $\omega^\omega$. In particular, every metrizable group is such, but the class…

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Z^n-trees give one a powerful tool to…

Recall that a group is called large if it has a finite index subgroup which surjects onto a non-abelian free group. By work of Agol and Cooper-Long-Reid, most 3-manifold groups are large; in particular, the fundamental groups of hyperbolic…

An $\omega$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $\omega$-tree-automatic structures. We prove…

For any proper action of a non-elementary group $G$ on a proper geodesic metric space, we show that if $G$ contains a contracting element, then there exists a sequence of proper quotient groups whose growth rate tends to the growth rate of…

This is the second paper in a series of three, where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Here, for an arbitrary group $G$ of infinite words over an ordered abelian group…

Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space the image of $p_#$, where $p_#: [X,E] -> [X,B]$. The isotropy subgroup of the map of…

In this paper we study the $ K $-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are $ K $-amenable and…