Finitely generated groups, \(p\)-adic analytic groups and Poincaré series (Q1801597)

The author is concerned with the connection of a group \(G\) with its Poincaré series \(\zeta_{G,p}(s) \equiv \sum^ \alpha_{n=0}a_{p^ n}(G)p^{-ns}\), where \(a_{p^ n}(G)\) is the number of subgroups of index \(p^ n\) in \(G\). Namely, the following question is considered in the paper: for which groups \(G\) and primes \(p\) can \(\zeta_{G,p}(s)\) be written as a rational function in \(p^{- s}\)? Rationality of Poincaré series associated with the following groups is proved: (i) For \(G\) a compact \(p\)-adic analytic group. (ii) For \(\Gamma\) a finitely generated group with finite upper rank. (iii) For \(\Gamma\) an arithmetic lattice inside \(G = \text{SL}_ n\), where \(n \geq 3\).