Exponential growth rates of free and amalgamated products

We prove that there is a gap between (sqrt 2 andleft( {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 (PGLleft( {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.