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Geometric generators for
braid-like groups
Daniel Allcock and Tathagata Basak |
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Geometry & Topology 20 (2016) 747–778
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Abstract |
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We study the problem of finding generators for the fundamental group of a space of the following sort: one removes a family of complex hyperplanes from , or complex hyperbolic space , or the Hermitian symmetric space for , and then takes the quotient by a discrete group . The classical example is the braid group, but there are many similar “braid-like” groups that arise in topology and algebraic geometry. Our main result is that if contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for in a particular intricate example in . The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group , that gives geometric meaning to the generators and relations in the Conway–Simons presentation of . We also suggest some other applications of our machinery. |
Keywords
fundamental groups, infinite hyperplane arrangement,
complex hyperbolic geometry, braid groups, Artin groups,
Leech lattice, presentations, Monster
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Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F36, 52C35, 32S22
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References |
Publication
Received: 10 March 2014
Revised: 21 April 2015
Accepted: 9 June 2015
Published: 28 April 2016
Proposed: Walter Neumann
Seconded: Peter Teichner, Ronald Stern
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Authors |
| Daniel Allcock | |
| Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin, TX 78712-1082 USA |
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| http://www.math.utexas.edu/~allcock | |
| Tathagata Basak | |
| Department of Mathematics Iowa State University Ames, IA 50011 USA |
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| http://orion.math.iastate.edu/tathagat | |
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