 
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 S1
}. 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
