The growth function of Coxeter dominoes and $2$–Salem numbers

Abstract

By the results of Cannon, Wagreich and Parry, it is known that the growth rate of a cocompact Coxeter group in ${ℍ}^2$ and ${ℍ}^3$ is a Salem number. Kerada defined a $j$–Salem number, which is a generalization of Salem numbers. In this paper, we realize infinitely many $2$–Salem numbers as the growth rates of cocompact Coxeter groups in ${ℍ}^4$. Our Coxeter polytopes are constructed by successive gluing of Coxeter polytopes, which we call Coxeter dominoes.