## 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.

### MSC:

 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

### Citations:

Zbl 1231.20027; Zbl 0735.51016
Full Text:

### References:

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