 
Summary: UNIFORM NONAMENABILITY
G.N. ARZHANTSEVA, J. BURILLO, M. LUSTIG, L. REEVES, H. SHORT,
AND E. VENTURA
Abstract. For any nitely generated group G an invariant Fl G
0 is introduced which measures the \amount of nonamenability"
of G. If G is amenable, then Fl G = 0. If Fl G > 0, we call G
uniformly nonamenable. We study the basic properties of this in
variant; for example, its behaviour when passing to subgroups and
quotients of G. We prove that the following classes of groups are
uniformly nonamenable: nonabelian free groups, nonelementary
wordhyperbolic groups, large groups, free Burnside groups of large
enough odd exponent, and groups acting acylindrically on a tree.
Uniform nonamenability implies uniform exponential growth. We
also exhibit a family of nonamenable groups (in particular in
cluding all nonsolvable BaumslagSolitar groups) which are not
uniformly nonamenable, that is, they satisfy Fl G = 0. Finally,
we derive a relation between our uniform Flner constant and the
uniform Kazhdan constant with respect to the left regular repre
sentation of G.
The work has been partially supported by the Swiss National Science Foundation
