 
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.
Introduction
