Event Detail

Event Type: 
Topology Seminar
Monday, November 11, 2013 - 04:00
Gilkey 115

Speaker Info

Boise State University

Let F/R be a presentation of a group G, and let R/[R,R] be the associated relation module. The relation lifting problem asks the question: given generators r1[R,R],…, rn[R,R] of the relation module R/[R,R], do there exist lifts r1c1,…, rncn  in R, where ci are in [R,R], that normally generate the relation group R? The relation lifting problem arose first in the work of Wall on finiteness obstructions in 1965. Dunwoody showed in 1972 that relations cannot always be lifted. His construction relied on the existence of non-trivial units in the group ring of Z5. Bestvina and Brady exhibited a presentation of a finitely generated torsion-free group that is not finitely presented but admits a finitely generated relation module. Thus, relation lifting fails in a very strong sense. A difference between the minimal number of relators and the minimal number of relation module generators is called the relation gap. Relation gaps in finite presentations have not been found so far, although there is no lack of examples where such a gap is expected to occur. In my talk I intend to survey aspects of the relation lifting and relation gap problem and present some new results.