7 research outputs found
The Baillon–Simons theorems
AbstractIn this paper, we give combinatorial proofs of Baillon and Simons’ almost fixed point and fixed point theorems for discrete-valued mappings (J. Combin. Theory Ser. A 60 (1992) 147–154)
Clique graphs and Helly graphs
AbstractAmong the graphs for which the system of cliques has the Helly property those are characterized which are clique-convergent to the one-vertex graph. These graphs, also known as the so-called absolute retracts of reflexive graphs, are the line graphs of conformal Helly hypergraphs possessing a certain elimination scheme. From particular classes of such hypergraphs one can readily construct various classes G of graphs such that each member of G has its clique graph in G and is itself the clique graph of some other member of G. Examples include the classes of strongly chordal graphs and Ptolemaic graphs, respectively
Helly groups
Helly graphs are graphs in which every family of pairwise intersecting balls
has a non-empty intersection. This is a classical and widely studied class of
graphs. In this article we focus on groups acting geometrically on Helly graphs
-- Helly groups. We provide numerous examples of such groups: all (Gromov)
hyperbolic, CAT(0) cubical, finitely presented graphical C(4)T(4) small
cancellation groups, and type-preserving uniform lattices in Euclidean
buildings of type are Helly; free products of Helly groups with
amalgamation over finite subgroups, graph products of Helly groups, some
diagram products of Helly groups, some right-angled graphs of Helly groups, and
quotients of Helly groups by finite normal subgroups are Helly. We show many
properties of Helly groups: biautomaticity, existence of finite dimensional
models for classifying spaces for proper actions, contractibility of asymptotic
cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones
conjecture and of the coarse Baum-Connes conjecture. This leads to new results
for some classical families of groups (e.g. for FC-type Artin groups) and to a
unified approach to results obtained earlier