Analytic properties of zeta functions and subgroup growth

In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a nilpotent group. In order to state our results we introduce the following notation. For \alpha a real number and N a nonnegative integer, define s_N^\alpha(G) = sum_{n=1}^N a_n(G)/n^\alpha. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence \alpha(G) of \zeta_G(s) is a rational number and \zeta_G(s) can be meromorphically continued to Re(s)>\alpha(G)-\delta for some \delta >0. The continued function is holomorphic on the line \Re(s) = (\alpha)G except for a pole at s=\alpha(G). (2) There exist a nonnegative integer b(G) and some real numbers c,c' such that s_{N}(G) ~ c N^{\alpha(G)}(\log N)^{b(G)} s_{N}^{\alpha(G)}(G) ~ c' (\log N)^{b(G)+1} for N\rightarrow \infty .Comment: 41 pages, published version, abstract added in migratio

On the Surjectivity of Engel Words on PSL(2,q)

We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2,q).Comment: v2: 25 pages, minor changes, accepted to the Journal of Groups, Geometry and Dynamic