Subgroup growth

Let $\Gamma$ be a finitely generated group. Denote by $\sigma_n(\Gamma)$ the number of subgroups of $\Gamma$ of index at most $n$. In the lecture we will survey connections between the algebraic structure of $\Gamma$ and the growth of $\sigma_n(\Gamma$)---the "subgroup growth of $\Gamma$".

The central themes will be:

1. The characterization of the groups with polynomial subgroup growth (PSG-groups).

This was established in a series of papers (cf. [LMS] and the references therein) using the classification of finite simple groups, the theories of solvable groups, $p$-adic Lie groups, algebraic groups, arithmetic groups and the prime number theorem.

2. Counting congruence subgroups and subgroup growth of linear groups.

In [L], the growth rate of the number of congruence subgroups of an arithmetic group was determined (it is $n^{c\log n/\log\log n}$) and it was shown that the congruence subgroup property (CSP) can be characterized by mens of subgroup growth. This gives a meaning to the CSP for non-arithmetic lattices in semi-simple Lie groups, e.g. fundamental groups of general hyperbolic manifolds. We also deduced a sharp lower bound on the subgroup growth of every non-PSG linear group. The proofs involve counting methods in finite groups (cf. [P]).

3. Groups of intermediate subgroup growth.

Arithmetic groups with CSP (e.g., $\Gamma={\rm SL}_3({\bf Z}))$ has intermediate subgroup growth as do other examples constructed in [SS] of groups with fractionally exponential growth.

4. Virtually free groups.

M. Hall calculated in 1949 $\sigma_n(\Gamma)$ for free groups $\Gamma$. Far reaching generalizations for virtually free groups, were achieved in [M] (and the references therein), using methods of combinatorial analysis and complex functions.

5. Connection with pro-finite groups $\sigma_n(G)$

when $G$ is a pro-finite group has been studied for its own sake as well as for applications to discrete groups. For example PSG-pro-$p$ groups are exactly the $p$-adic analytic pro-$p$-groups ([LM], [DDMS]) and counting maximal subgroups of $G$ turns out to be related to

6. Zeta function of groups.

Let $\zeta_\Gamma(s)=\sum a_n(\Gamma)n^{-s}$ be the ``zeta function of $\Gamma^n$ where $a_n(\Gamma)=\sigma_n(\Gamma)-\sigma_{n-1}(\Gamma)$. Rationality results for Euler factors of $\zeta_\Gamma(s)$ were proved under the assumption that $\Gamma$ is nilpotent ([GSS]) (see also [DDMS], Appendix C). Connections with buildings and Igusa's local zeta function function enable us to deduce some functional equations ([dSL]).

Bibliography:

1. [DDMS] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-$p$ Groups , LMS Lecture Notes Series 157, Cambridge Univ. Press, 1991.
2. [dSL] M. P. F. du Sautoy, A. Lubotzky, Functional equations and uniformity for local zeta functions of nilpotent groups , preprint.
3. [GSS] F. J. Grunewald, D. Segal and G. C. Smith, Subgroups of finite index in nilpotent groups , Invent. Math. 93 (1988), 185--223.
4. [L] A. Lubotzky, Subgroup growth and congruence subgroups , Invent. Math., to appear.
5. [LM] A Lubotzky, A. Mann, On groups of polynomial subgroup growth , Invent. Math. 104 (1991), 521--553.
6. [LMS] A. Lubotzky, A. Mann and D. Segal, Finitely generated groups of polynomial subgroup growth , Israel J. Math. 82 (1993), 363--371.
7. [MS] A. Mann, A. Shalev, Maximal subgroups of finite simple groups and positively finitely generated groups , preprint.
8. [M] T. Müller, Finite group actions, subgroups of finite index in free products and asymptotic expansion of $\epsilon^{p(z)$ , preprint.
9. [P] L. Pyber, Enumerating finite groups of given order , Ann. of Math. 137 (1993), 203--220.
10. [SS] D. Segal, A. Shalev, Groups of fractionally exponential subgroup growth, J. of Pure and App. Alg. 88 (1993), 205--223.