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UNIFORM NON-AMENABILITY G.N. ARZHANTSEVA, J. BURILLO, M. LUSTIG, L. REEVES, H. SHORT,
 

Summary: UNIFORM NON-AMENABILITY
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 non-amenability"
of G. If G is amenable, then Fl G = 0. If Fl G > 0, we call G
uniformly non-amenable. 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 non-amenable: non-abelian free groups, non-elementary
word-hyperbolic groups, large groups, free Burnside groups of large
enough odd exponent, and groups acting acylindrically on a tree.
Uniform non-amenability implies uniform exponential growth. We
also exhibit a family of non-amenable groups (in particular in-
cluding all non-solvable Baumslag-Solitar groups) which are not
uniformly non-amenable, 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

  

Source: Arzhantseva, Goulnara N. - Section de Mathématiques, Université de Genève

 

Collections: Mathematics