Finite index subgroups of arrangement groups
by
Daniel Matei
Northeastern University
Coauthors: Alexander I. Suciu (Northeastern University)

Let G be a finitely presented group, and p a prime number. A natural invariant of G is the number N_p(G) = #{ K \lhd G | [G:K]=p } of index p normal subgroups. We introduce a series of numerical invariants of G by counting the index p normal subgroups K of G according to

• their first Betti number:
  

  \betap, d(G) = #{ K  | b1(K) = b1(G) + (p-1) d  };

• the q-torsion of their abelianization, for a prime q =/= p:
  

  \gammap, q, d(G) = #{ K   | dim ZqTors H1(K)\otimesZq = (p-1) d + dim ZqTors H1(G)\otimesZq  } ;

• the p-torsion of their abelianization:
  

  \nup, d(G) = #{ K   | dim ZpTors H1(K)\otimesZp = d  }.

In this talk we discuss the above invariants when G is either the fundamental group of the complement of an arrangement, or a nilpotent quotient of such a group. We consider both complex line arrangements in C² and real plane arrangements in R⁴.

The central idea we explore is that the numbers N_p(G), \beta_{p, d}(G), \gamma_{p, q, d}(G), and \nu_{p, d}(G) are determined by certain algebraic varieties (over fields of appropriate characteristic) associated to the group G: the characteristic varieties and the resonance varieties.

Date received: June 7, 1999