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Free products with amalgamation and HNN-extensions of uniformly exponential growth. (English. Russian original) Zbl 0998.20025

Math. Notes 67, No. 6, 686-689 (2000); translation from Mat. Zametki 67, No. 6, 811-815 (2000).
A finitely generated group has exponential growth, with respect to finite generating set \(S\), if the number of elements of \(G\) having word representatives of length \(\leq n\) in the letters of \(S\) grows asymptotically at a rate \(r>1\). The same group has uniformly exponential growth if \(r\) can be chosen independently of \(S\).
M. Gromov has asked for an example of a finitely-generated group of exponential growth which is not of uniformly exponential growth. The authors limit the manner in which such an example might conceivably be constructed by proving the following two results: (1) A free product \(A*_CB\) with amalgamation has uniformly exponential growth provided that \(A\) and \(B\) are finitely-generated groups with common subgroup \(C\) such that \(([A:C]-1)([B:C]-1)\geq 2\).
(2) An HNN extension \(G*_{C_1=C_2}\) has uniformly exponential growth provided that \(G\) is finitely generated and \(C_1\) and \(C_2\) are isomorphic subgroups such that \([G:C_1]+[G:C_2]\geq 3\). – The proofs involve actions on trees.
Reviewer: J.W.Cannon (Provo)

MSC:

20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F69 Asymptotic properties of groups
20E08 Groups acting on trees
20F05 Generators, relations, and presentations of groups
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