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Geometric generators for braid-like groups

Daniel Allcock and Tathagata Basak

Geometry & Topology 20 (2016) 747–778

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n, or complex hyperbolic space n, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group PΓ. 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 PΓ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then G is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for G in a particular intricate example in 13. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group M, that gives geometric meaning to the generators and relations in the Conway–Simons presentation of (M × M) : 2. We also suggest some other applications of our machinery.

fundamental groups, infinite hyperplane arrangement, complex hyperbolic geometry, braid groups, Artin groups, Leech lattice, presentations, Monster
Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F36, 52C35, 32S22
Received: 10 March 2014
Revised: 21 April 2015
Accepted: 9 June 2015
Published: 28 April 2016
Proposed: Walter Neumann
Seconded: Peter Teichner, Ronald Stern
Daniel Allcock
Department of Mathematics
University of Texas at Austin
1 University Station C1200
Austin, TX 78712-1082
Tathagata Basak
Department of Mathematics
Iowa State University
Ames, IA 50011