Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
GROWTH RATES OF AMENABLE GROUPS G. N. ARZHANTSEVA, V. S. GUBA, AND L. GUYOT
 

Summary: GROWTH RATES OF AMENABLE GROUPS
G. N. ARZHANTSEVA, V. S. GUBA, AND L. GUYOT
Abstract. Let Fm be a free group with m generators and let R
be its normal subgroup such that Fm/R projects onto Z. We give
a lower bound for the growth rate of the group Fm/R # (where R # is
the derived subgroup of R) in terms of the length # = #(R) of the
shortest non­trivial relation in R. It follows that the growth rate
of Fm/R # approaches 2m-1 as # approaches infinity. This implies
that the growth rate of an m­generated amenable group can be
arbitrarily close to the maximum value 2m - 1. This answers an
open question of P. de la Harpe. We prove that such groups can
be found in the class of abelian­by­nilpotent groups as well as in
the class of virtually metabelian groups.
1. Introduction
Let G be a finitely generated group and A a fixed finite set of gener­
ators for G. We denote by #(g) the word length of an element g # G in
the generators A, i.e. the length of a shortest word in the alphabet A ±1
representing g. Let B(n) denote the ball {g # G | #(g) # n} of radius n
in G with respect to A. The growth rate of the pair (G, A) is the limit
#(G, A) = lim

  

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

 

Collections: Mathematics