 
Summary: Geometry & Topology 13 (2009) 12291263 1229
Infinite groups with fixed point properties
GOULNARA ARZHANTSEVA
MARTIN R BRIDSON
TADEUSZ JANUSZKIEWICZ
IAN J LEARY
ASHOT MINASYAN
JACEK ´SWIA¸ TKOWSKI
We construct finitely generated groups with strong fixed point properties. Let Xac
be the class of Hausdorff spaces of finite covering dimension which are modp
acyclic for at least one prime p. We produce the first examples of infinite finitely
generated groups Q with the property that for any action of Q on any X 2 Xac ,
there is a global fixed point. Moreover, Q may be chosen to be simple and to have
Kazhdan's property (T). We construct a finitely presented infinite group P that admits
no nontrivial action on any manifold in Xac . In building Q, we exhibit new families
of hyperbolic groups: for each n 1 and each prime p, we construct a nonelementary
hyperbolic group Gn;p which has a generating set of size nC2, any proper subset of
which generates a finite pgroup.
20F65, 20F67; 57S30, 55M20
Dedicated to Michael W Davis on the occasion of his 60th birthday
