Growth tightness for word hyperbolic groups

Abstract

Abstract. We show that non-elementary 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.

A Lower Bound on the Growth of Word Hyperbolic Groups

“…However, even for this case, the question seems to be highly non-trivial. In particular, as observed in [1], an affirmative answer to the question would imply that a non-elementary word hyperbolic group be Hopfian. Note that word hyperbolic groups that are torsion free are known to be Hopfian [14].…”

Growth and ergodicity of context-free languages

“…Indeed, it is clearly enough to prove this when F = {v 1 } consists of a single word v 1 . If m = |v 1 |, then after m − 1 iterations, v m = φ (m−1) (v 1 ) is a letter in the alphabet of φ (m−1) (L), and we have…”

“…However, there is no general theory regarding growth-sensitive automatic groups. Arzhantseva and Lysenok [1] have recently proved that every non-elementary word-hyperbolic group has a growth-sensitive, regular geodesic normal form. For a specific example, where other related languages are also studied, see Bartholdi and Ceccherini-Silberstein [2].…”

“…This tour has a i as its first label. After that, there are three possibilities: (1) we have reached h, which is a leaf, and we produce some element of Nf K (at the terminal point it may be = ε), (2) we have reached h, produce an element of Nf K and perform additional loops around some of the branches emanating from h, (3) we have not yet reached h, produce an element of Nf K , perform loops around some of the branches that do not contain h and proceed further towards h. Now suppose that Nf K is regular and that C K is an unambiguous, left or right linear grammar. In the dependency di-graph associated with our grammar C for Nf G , we see that S is isolated, and the other strong components are V T = {T 1 , .…”

Growth exponent of generic groups

“…Note however that this exponent cannot stay exactly the same, as Arzhantseva and Lysenok proved in [AL02] that quotienting a hyperbolic group by an infinite normal subgroup decreases the growth exponent.…”