Exponential growth rates of free and amalgamated products. (English) Zbl 1350.20020

Summary: We prove that there is a gap between \(\sqrt 2\) and \((1+\sqrt 5)/2\) for the exponential growth rate of nontrivial free products. For amalgamated products \(G=A*_CB\) with \(([A:C]-1)([B:C]-1)\geq 2\), we show that an exponential growth rate lower than \(\sqrt 2 \) can be achieved. Indeed, there are infinitely many amalgamated products for which the exponential growth rate is equal to \(\psi\approx 1.325\), where \(\psi\) is the unique positive root of the polynomial \(z^3-z-1\). One of these groups is \(\mathrm{PGL}(2,\mathbb Z)\cong(C_2\times C_2)*_{C_2}D_6\). However, under some natural conditions the lower bound can be put up to \(\sqrt 2\). This answers two questions by A. Mann [J. Algebra 326, No. 1, 208-217 (2011; Zbl 1231.20027)]. We also prove that \(\psi\) is a lower bound for the minimal growth rates of a large class of Coxeter groups, including cofinite non-cocompact planar hyperbolic groups, which strengthens a result obtained earlier by William Floyd, who considered only standard Coxeter generators.


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