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UNIFORM GROWTH, ACTIONS ON TREES AND GL2
 

Summary: UNIFORM GROWTH, ACTIONS ON
TREES AND GL2
Roger C. Alperin and Guennadi A. Noskov
1. Exponential Growth
Choose a finite generating set S = {s1, , sp} for the group ; de-
fine the S -length of an element as S(g) = min{ n | g = s1 sn, si
S S-1
}. The growth function n(S, ) = |{g | S(g) n}| depends
on the chosen generating set. A group has exponential growth if the
growth rate, (S, ) = limitnn(S, )
1
n is strictly greater than 1.
In fact, for another finite generating set T = {t1, , tq} for , if both
maxjS(tj) L and maxiT (si) L, then n(S, ) Ln(T, ) and
also the symmetric inequality. It then follows that (S, )L
(T, )
and (T, )L
(S, ). Using these remarks, Milnor showed that
exponential growth is independent of the generating set.
For a group with exponential growth we consider

  

Source: Alperin, Roger C. - Department of Mathematics, San Jose State University
Bielefeld, University of - Spectral Analysis, Asymptotic Distributions and Stochastic Dynamic Group

 

Collections: Mathematics