The edge zeta function of the standard non-uniform PGL3 quotient is computed exactly as a rational function, yielding an exact count of closed geodesic cycles.
Equidistribution and counting under equilibrium states in negative curvature and trees. Applications to non-Archimedean Diophantine approximation
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abstract
In this book, we study equidistribution and counting problems concerning locally geodesic arcs in negatively curved spaces endowed with potentials, and we deduce, from the special case of tree quotients, various arithmetic applications to equidistribution and counting problems in non-Archimedean local fields.
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Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$
The edge zeta function of the standard non-uniform PGL3 quotient is computed exactly as a rational function, yielding an exact count of closed geodesic cycles.