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On the growth of groups and automorphisms

#### Martin R. Bridson

#### To appear in IJAC volume in honour of R.Grigorchuk

We consider the growth functions $\beta_\G(n)$ of amalgamated
free products $\G=A\ast_C B$, where $A\cong B$ are finitely generated, $C$
is free abelian and $|A/C|=|A/B|=2$. For every $d\in\mathbb N$
there exist examples with $\beta_\G(n)\simeq n^{d+1}\beta_A(n)$. There
also exist examples with $\beta_\Gamma(n)\simeq e^n$. Similar behaviour
is exhibited among Dehn functions.