Totally chiral maps and hypermaps of small genus
Totally chiral maps and hypermaps of small genus
An orientably regular hypermap is totally chiral if it and its mirror image have no non-trivial common quotients. We classify the totally chiral hypermaps of genus up to 1001, and prove that the least genus of any totally chiral hypermap is 211, attained by twelve orientably regular hypermaps with monodromy group A7 and type (3,4,4) (up to triality). The least genus of any totally chiral map is 631, attained by a chiral pair of orientably regular maps of type {11,4}, together with their duals; their monodromy group is the Mathieu group M11. This is also the least genus of any totally chiral hypermap with non-simple monodromy group, in this case the perfect triple covering 3.A7 of A7. The least genus of any totally chiral map with non-simple monodromy group is 1457, attained by 48 maps with monodromy group isomorphic to the central extension 2.Sz(8).
map, hypermap, totally chiral map, totally chiral hypermap, chirality group, 2-generated groups
3971-3996
Breda d'Azevedo, Antonio
ea38b7ac-138c-4cc7-97fb-fd271d4243cf
Jones, Gareth
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
1 December 2009
Breda d'Azevedo, Antonio
ea38b7ac-138c-4cc7-97fb-fd271d4243cf
Jones, Gareth
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Abstract
An orientably regular hypermap is totally chiral if it and its mirror image have no non-trivial common quotients. We classify the totally chiral hypermaps of genus up to 1001, and prove that the least genus of any totally chiral hypermap is 211, attained by twelve orientably regular hypermaps with monodromy group A7 and type (3,4,4) (up to triality). The least genus of any totally chiral map is 631, attained by a chiral pair of orientably regular maps of type {11,4}, together with their duals; their monodromy group is the Mathieu group M11. This is also the least genus of any totally chiral hypermap with non-simple monodromy group, in this case the perfect triple covering 3.A7 of A7. The least genus of any totally chiral map with non-simple monodromy group is 1457, attained by 48 maps with monodromy group isomorphic to the central extension 2.Sz(8).
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Published date: 1 December 2009
Keywords:
map, hypermap, totally chiral map, totally chiral hypermap, chirality group, 2-generated groups
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Local EPrints ID: 156471
URI: http://eprints.soton.ac.uk/id/eprint/156471
ISSN: 0021-8693
PURE UUID: 42e197ab-0c25-4275-b8ea-1787abdbff6b
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Date deposited: 01 Jun 2010 09:42
Last modified: 14 Mar 2024 01:43
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Author:
Antonio Breda d'Azevedo
Author:
Gareth Jones
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