Seminar of Algebra

For a pair $(G, S)$ of a group $G$ and a finite generating set $S$, the growth rate $\tau(G, S)$ measures how fast the number of elements of $G$ increases in the word length with respect to $S$. The growth rates of the Coxeter systems in hyperbolic geometry have been investigated, focusing on the relationship between the arithmetic properties of the growth and the geometric properties of the fundamental polyhedron. In this talk, we first give an overview of previous studies on the growth rates of Coxeter systems in hyperbolic geometry. After that, we provide generalizations of the known results to general Coxeter system.