Bertrand's postulate and subgroup growth.
Wed, 09/09/2009 - 22:08 — SomNambul
link: http://arxiv.org/abs/0909.1343
Abstract
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.
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