Exponential growth rates of free and amalgamated products

We prove that there is a gap between for the exponential growth rate of nontrivial free products. For amalgamated products G = A (*C) B with ([A: C] - 1)([B: C] - 1) a parts per thousand yen 2, we show that an exponential growth rate lower than can be achieved. Indeed, there are infinitely many amalgamated products for which the exponential growth rate is equal to psi a parts per thousand 1.325, where psi is the unique positive root of the polynomial z (3)-z-1. One of these groups is . However, under some natural conditions the lower bound can be put up to . 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 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.