Two-generator one-relator groups and marked polytopes

2015/01/14 by Stefan Friedl, Stephan Tillmann, Friedl, Stefan +1 · 2 citations
Mathematics · #20F65 #20J05 #22E40 #57R19 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F65 #msc:20J05 #msc:22E40 #msc:57R19

paper · pdf · doi:10.48550/arxiv.1501.03489

33 pages, 5 figures

arxiv created 2015/01/14 · arxiv updated 2015/01/15

Abstract

We use Fox calculus to assign a marked polytope to a `nice' group presentation with two generators and one relator. Relating the marked vertices to Novikov-Sikorav homology we show that they determine the Bieri-Neumann-Strebel invariant of the group. Furthermore we show that in many cases the marked polytope is an invariant of the underlying group and that in those cases the marked polytope also determines the minimal complexity of all the associated HNN-splittings.

Cited by

• Universal L²-torsion and sutured decomposition for 3-manifolds
• The word problem for two-generator one-relator inverse monoids

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