Summary: GROWTH TIGHTNESS FOR WORD HYPERBOLIC GROUPS
G. N. ARZHANTSEVA AND I. G. LYSENOK
Abstract. We show that nonelementary word hyperbolic groups are growth
tight. This means that, given such a group G and a finite set A of its generators,
for any infinite normal subgroup N of G, the exponential growth rate of G=N with
respect to the natural image of A is strictly less than the exponential growth rate
of G with respect to A.
Let G be a finitely generated group and A a finite set of generators for G. By
jxj we denote the geodesic length of an element x 2 G in the generators A, i.e. the
length of a shortest word in the alphabet A \Sigma1 representing x. Let B(n) denote the
ball fg 2 G j jgj Ÿ ng of radius n in G.
The exponential growth rate of the pair (G; A) is the limit
–(G; A) = lim
where #X denotes the number of elements of a finite set X. The existence of the
limit follows from the submultiplicativity property of the function #B(n): #B(m+