Bertrand's postulate and subgroup growth

TL;DR

This paper introduces the concept of residual average in finitely generated groups, linking subgroup growth rates to prime gap analogies, and proves its finiteness for linear groups.

Contribution

It defines residual average using divisibility functions and establishes its finiteness for finitely generated linear groups, connecting subgroup growth to prime gap phenomena.

Findings

Residual averages are finite for finitely generated linear groups.

Growth rates of subgroup indices relate to prime gap analogies.

Provides a new perspective on subgroup growth functions.

Abstract

In this article we investigate the L^1-norm of certain functions on groups called divisibility functions. Using these functions, their connection to residual finiteness, and integration theory on profinite groups, we define the residual average of a finitely generated group. One of the main results in this article is the finiteness of residual averages on finitely generated linear groups. Whether or not the residual average is finite depends on growth rates of indices of finite index subgroups. Our results on index growth rates are analogous to results on gaps between primes, and provide a variant of the subgroup growth function, which may be of independent interest.