2 research outputs found
Word images in symmetric and classical groups of Lie type are dense
Let be a non-trivial 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 distance resp. the normalized rank metric on when
is a symmetric group resp. one of the other classical groups and write
for the permutation resp. Lie rank 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 a conjectures by Shalev and
Larsen. Equivalently, we prove that any non-trivial 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-Shalev that for non-trivial words and
sufficiently large.Comment: 28 pages, no figure