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Abstract
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Let
be a nontrivial word
and denote by
the image of
the associated word map
.
Let
be one of the
finite groups
( a prime
power,
,
), or the
unitary group
over
.
Let
be the normalized Hamming metric resp. the normalized rank metric on
when
is a symmetric group resp. one of the other classical groups and write
for the
degree of
.
For
, we prove that
there exists an integer
such that
is
-dense in
with respect
to the metric
if
.
This confirms metric versions of conjectures by Shalev and Larsen. Equivalently, we
prove that any nontrivial word map is surjective on a metric ultraproduct of groups
from above
such that
along the ultrafilter.
As a consequence of our methods, we also obtain an
alternative proof of the result of Hui, Larsen, and Shalev that
for nontrivial
words
and
sufficiently large.
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Keywords
word map, word image, finite groups of Lie type, symmetric
groups, Hamming metric, cohomology
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Mathematical Subject Classification 2010
Primary: 20B30, 20D06, 20H30
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Milestones
Received: 13 November 2019
Revised: 2 April 2020
Accepted: 19 May 2020
Published: 31 July 2021
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