Related papers: Sofic actions on graphs
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular…
We undertake a systematic study of the approximation properties of the topological and measurable versions of the coarse boundary groupoid associated to a sequence of finite graphs of bounded degree. On the topological side, we prove that…
I define a "group graph" which encodes the symmetry in a dynamical process on a network. Group graphs extend signed networks, where links are labelled with plus or minus one, by allowing link labels from any group and generalising the…
Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which…
We define and study notions of amenability and skew-amenability of continuous actions of topological groups on compact topological spaces. Our main motivation is the question under what conditions amenability of a topological group passes…
We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we…
In recent work, Stokes and Vermant considered graph-of-groups realisations of hypergraphs as a new description of rigidity-theoretic problems. In this paper, we show that the infinitesimal aspects of graph-of-groups realisations can be…
We show that the class of sofic actions is closed under direct products and contains a (non-unique) maximal element in the weak containment order. For any sofic group we construct nice sofic approximations such that all the sofic actions…
When one studies geometric properties of graphs, local finiteness is a common implicit assumption, and that of transitivity a frequent explicit one. By compactness arguments, local finiteness guarantees several regularity properties. It is…
We consider a finite, connected and simple graph $\Gamma$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(\Gamma)$, the local action $G_x^{\Gamma(x)}$ is the action of…
We establish a strong, geometric lower bound on the (sequential) topological complexity of the unordered configuration spaces of a general graph. As an application, we show that, for most graphs, the topological complexity eventually…
Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the…
The class A of countable groups that admit a faithful, transitive, amenable -- in the sense that there is an invariant mean -- action on a set has been widely investigated in the past. In this paper, we no longer require the action to be…
We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above…
We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original…
Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and…
We consider a finitely generated group acting minimally on a compact space by homeomorphsims, and assume that the Schreier graph of at least one orbit is quasi-isometric to a line. We show that the topological full group of such an action…
Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local…
A Cayley Graph for a group $G$ is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the Holomorph of $G$ (the normaliser of a regular copy of $G$ in $\operatorname{Sym}(G)$). We complete the…
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…