Alexander Kolpakov, Vanderbilt University
“Growth rates of Coxeter groups, tessellations of hyperbolic space and algebraic integers”
In this I will first introduce the growth rate of a Coxeter group. This is a number associated to such a group, which turns later on to be a nice algebraic integer (Salem or Pisot number) if the group acts on the hyperbolic space of dimension n = 2 or 3. The convergence of Salem numbers to Pisot numbers, well-known in number theory, has a geometric representation in the hyperbolic plane (dimension n = 2), following the work by W. Floyd and W. Parry. Then I shall discuss this geometric interpretation in the hyperbolic space of dimension n = 3 and speak about some analogies to higher dimensions n ≥ 4. Also we will see what are the minimal growth rates of Coxeter groups acting on the hyperbolic space of dimension n = 2 (result by E. Hironaka) and n = 3 (joint work by Ruth Kellerhals and A.K.).