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Free by cyclic groups and linear groups with restricted unipotent elements

Free by cyclic groups and linear groups with restricted unipotent elements AbstractWe introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic.We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6].We give examples of abstract groups lying in this class, but also show that Gersten’sfree by cyclic group does not.This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Free by cyclic groups and linear groups with restricted unipotent elements

Button, Jack O.
Groups Complexity Cryptology , Volume 9 (2): 13 – Nov 1, 2017
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Publisher
de Gruyter
Copyright
© 2017 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6104
eISSN
1869-6104
DOI
10.1515/gcc-2017-0009
Publisher site
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Abstract

AbstractWe introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic.We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6].We give examples of abstract groups lying in this class, but also show that Gersten’sfree by cyclic group does not.This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group.

Journal

Groups Complexity Cryptology – de Gruyter

Published: Nov 1, 2017

References

  • Generalized free products of linear groups
  • Braid groups are linear
  • Braid groups are linear
  • A 3-manifold group which is not four dimensional linear
  • The automorphism group of a free group is not a CAT⁢(0){\rm CAT}(0) group