Related papers: Shortcut Graphs and Groups

We show that whether loops can be shortcut in a group's Cayley graph depends on the choice of finite generating set. Our example is the direct product of two rank-2 free groups and a consequence is that this group has asymptotic cones with…

We define the strong shortcut property for rough geodesic metric spaces, generalizing the notion of strongly shortcut graphs. We show that the strong shortcut property is a rough similarity invariant. We give several new characterizations…

We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every…

We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with…

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges…

We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each…

We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete.…

In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple class of quasi-isometrically rigid subgroups in graph products of finite groups, which we call eccentric subgroups. As an application, we prove that,…

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 define a family of graphs we call dual systolic graphs. This definition comes from graphs that are duals of systolic simplicial complexes. Our main result is a sharp (up to constants) isoperimetric inequality for dual systolic graphs.…

We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is…

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some…

Shortcut sets are a vital instrument for reducing the diameter of a static graph and, consequently, its shortest path complexity, which is relevant in numerous subfields of graph theory. We explore the notion of shortcut sets in temporal…

The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the…

In this paper, we investigate the graphs in which all balls are convex and the groups acting on them geometrically (which we call CB-graphs and CB-groups). These graphs have been introduced and characterized by Soltan and Chepoi (1983) and…

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups…

A subperiodic group is a group of motions of $d$-dimensional Euclidean space $\R^d$ which contains a translation lattice $\Z^r$ of rank $r < d$ as a subgroup of finite index. A classification into abstract group isomorphism classes is…

We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual…