On the set of critical exponents of discrete groups acting on regular trees

J. Korean Math. Soc. 2019 Vol. 56, No. 2, 475-484 Published online 2019 Mar 01

Sanghoon Kwon Catholic Kwandong University

Abstract : We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number $\delta$ between $0$ and $\frac{1}{2}\log q$, there is a discrete subgroup $\Gamma$ acting without inversion on a $(q+1)$-regular tree whose critical exponent is equal to $\delta$. Explicit construction of edge-indexed graphs corresponding to a quotient graph of groups are given.

Keywords : groups acting on trees, critical exponents, Ihara zeta function