Lifting representations of Z-groups

2004/05/24 by Daniel S. Silver, Susan G. Williams, Susan Williams +2
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GR #math.GT #msc:20E07 #msc:37B10 #msc:57M27

paper · pdf · doi:10.48550/arxiv.math/0405457

Version 2 has a new example 3.6 and other small revisions. To appear in Israel J. Math. Plain TeX, 14 pages with 1 eps figure

arxiv created 2005/03/04 · arxiv updated 2009/12/01

Abstract

Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S₂ lifts to a homomorphism onto S₃, and any homomorphism from K onto Z₃ lifts to a homomorphism onto the alternating group A₄.

Related

• Knot Group Epimorphisms
• Counting homomorphisms onto finite solvable groups
• Formulating basic notions of finite group theory via the lifting\n property
• Lifting Generators in Connected Lie Groups
• The unreasonable power of the lifting property in elementary mathematics