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Abstract
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For
a finitely
generated group and
,
we say
is detected by
a normal subgroup
if
. The
depth
of
is the lowest index of a normal, finite index subgroup
that detects
. In this paper we study
the expected depth,
,
where
is a
random walk on
.
We give several criteria that imply that
where
is the intersection of all normal subgroups of index at most
.
In particular, the equality holds in the class of all nilpotent groups
and in the class of all linear groups satisfying Kazhdan’s property
. We
explain how the right-hand side above appears as a natural limit and also give an
example where the convergence does not hold.
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Keywords
expectation, residual finiteness growth, intersection
growth, residual averages, depth function
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Mathematical Subject Classification 2010
Primary: 05C81, 20F65, 60B15
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Milestones
Received: 8 November 2017
Accepted: 10 July 2018
Published: 8 March 2019
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