Related papers: A combination theorem for affine tree-free groups

We prove in ZFC that an abelian group $C$ is cotorsion if and only if $\operatorname{Ext}(F,C) = 0$ for every $\aleph_k$-free group $F$, and discuss some consequences and related results. This short note includes a condensed overview of the…

We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an…

The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether the map $\hat{Aut\left(\Gamma\right)}\to Aut(\hat{\Gamma})$ is injective, or more generally, what is its kernel $C\left(\Gamma\right)$? Here $\hat{X}$…

For any countable group with infinite conjugacy classes we construct a family of forests on the group. For each of them there is a random walk on the group with the property that its sample paths almost surely converge to the geometric…

Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy…

Let $\rho: \Gamma \to G$ be an Anosov representation, with $\Gamma$ a word hyperbolic group and $G$ a semisimple Lie group. Previous works (Guichard--Wienhard, Kapovich--Leeb--Porti, and Carvajales--Stecker) constructed an open domain of…

For $\lambda>1$, we consider the locally free ${\mathbb Z}\ltimes_\lambda{\mathbb R}$ actions on ${\mathbb T}^2$. We show that, if the action is $C^r$ with $r\geq2$, then it is $C^{r-\epsilon}$-conjugate to an affine action generated by a…

We introduce a notion of shadowing property for actions of finitely generated groups and study its basic properties. We formulate and prove a shadowing lemma for actions of nilpotent groups. We construct an example of a faithful linear…

We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then…

Let $A$ be a set and $f:A\rightarrow A$ a bijective function. Necessary and sufficient conditions on $f$ are determined which makes it possible to endow $A$ with a binary operation $*$ such that $(A,*)$ is a cyclic group and $f\in…

In this note we generalise some of the work of Klarner on free semigroups of affine maps acting on the real line by using a classical approach from geometric group theory (the Ping-Pong lemma). We also investigate the boundaries within…

Consider a smooth action $\mathbf G\times M \rightarrow M$ of a compact connected Lie group $\mathbf G$ on a connected manifold $M$. Assume the existence of a point of $M$ whose isotropy group has a single element (free point). Then we…

We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups.

It is proved that an arbitrary finite group acting locally linearly, homologically trivially, and pseudofreely on a closed, simply connected 4-manifold must in fact be cyclic and act semifreely, provided the second betti number of the…

We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a {\em tree algebra}. Using the Riemann-Hilbert correspondence, we…

We investigate translation length functions for two-generated groups acting by isometries on $\Lambda$-trees, where $\Lambda$ is a totally ordered abelian group. In this context, we provide an explicit formula for the translation length of…

We prove several results on the model theory of Artin groups, focusing on Artin groups which are ``far from right-angled Artin groups''. The first result is that if $\mathcal{C}$ is a class of Artin groups whose irreducible components are…

We present some Zermelo-Fraenkel consistency results regarding bi-orderability of groups, as well as a construction of groups with Conradian orders whose every action on metric spaces has bounded orbits. A classical consequence of the…

We characterize rational actions of the additive group on algebraic varieties defined over a field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical…

Let G be a real semisimple Lie group with no compact factors and finite centre, and let $\Lambda$ be a lattice in G. Suppose that there exists a homomorphism from $\Lambda$ to the outer automorphism group of a right-angled Artin group…