Exponential growth rates of free and amalgamated products

We prove that there is a gap between \(\sqrt 2 and\left( {1 + \sqrt 5 } \right)/2\) for the exponential growth rate of nontrivial free products. For amalgamated products G = A_*CB with (A: C] ? 1)(B: C] ? 1) ≥ 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 ψ ≈ 1.325, where ψ is the unique positive root of the polynomial z³?z?1. One of these groups is \(PGL\left( {2,\mathbb{Z}} \right) \cong \left( {{C_2} \times {C_2}} \right){*_{{C_2}}}{D_6}\). However, under some natural conditions the lower bound can be put up to \(\sqrt 2 \). This answers two questions by Avinoam Mann The growth of free products, Journal of Algebra 326, no. 1 (2011), 208–217]. We also prove that ψ 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.