Geometric generators for braid-like groups

Abstract

Abstract. 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 C n , or complex hyperbolic space CH n , or the Hermitian symmetric space for O(2, n), and then takes the quotient by a discrete group P Γ. 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 P Γ 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 CH 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.

Locally toroidal polytopes of rank 6 and sporadic groups

“…We explain below that on the other hand the identification of the group of Γ 1 does not need coset enumeration, and can be carried out by hand or with any software capable of multiplying 24 × 24 matrices over F 2 , the field of 2 elements. It is worth mentioning that both F and the facet stabiliser (2 2 .4 2 ):F 4 are maximal subgroups in Ω, although only F remains maximal upon restricting to the simple subgroup O + 8 (2) of Ω. In Section 3 we sketch how to establish the existence of the group of Γ 1 as a subgroup in Fi 22 .…”

“…e.g. [24,8,14,3,1,2]. For instance, the Groups of this kind arise in the study of abstract regular polytopes, for which the book [17] by P. McMullen and E. Schulte is a definite reference.…”

“…In more detail, the facets {3, 3, 4, 3} s and vertex figures {3, 4, 3, 3} t here correspond to nontrivial finite quotients [3,3,4,3] s and [3,4,3,3] t of the affine Coxeter group F4 , i.e. the groups [3,3,4,3] and [3,4,3,3], with normal Abelian subgroups either of the form q 4 or q 2 × (2q) 2 , with q ≥ 2, where we follow notation from [6] to denote the direct product (Z/sZ) k of k copies of the cyclic group of order s by s k . The case q 4 is denoted in [16,17] by {3, 3, 4, 3} (q,0,0,0) , and the case q 2 × (2q) 2 by {3, 3, 4, 3} (q,q,0,0) (and completely similarly for {3, 4, 3, 3}).…”

“…Preliminary experiments with enumerating cosets of one of our finite [3 2 , 4, 3 2 ]quotients, namely the group of Γ 1 , isomorphic to O + 8 (2):S 3 , in [3 2 , 4, 3 3 ], indicates that this gives the group F 4 (2), which is a subgroup of a quotient of Y 333 isomorphic to 2 E 6 (2). More precisely, the enumeration returns index 12673024, which is 4 times the index of O + 8 (2):S 3 in F 4 (2). We should also mention that the [3, 4, 3 3 ]-subgroup in this group, which corresponds to a locally toroidal polytope of the type considered in [17,Sect.…”

Noncrossing partitions and Milnor fibers

“…Then g restricts to a locally trivial fibration g : C n − V → C * , the global Milnor fibration, with fiber g −1 (1), the Milnor fiber of g [23]. The topology of g −1 (1) and the monodromy of the bundle are invariants of the singularity type of g at the origin. Of special interest is the case where g = Q W is a product of complex linear forms defining the arrangement A = A W of reflecting hyperplanes in C n of a finite real or complex reflection group W , see [25,26,21,17].…”