UCL

### Negative immersions and coherence of (most) one-relator groups

Geometry and Topology Seminar

17th May 2022, 2:00 pm – 3:00 pm

Fry Building,
2.04

One-relator groups G=F/<> as a class are something of an outlier in geometric group theory. On the one hand they have some good algorithmic properties, e.g. solvable word problem, but pathological examples abound, and they have therefore been resistant to most of the geometric tools we have available. I will relate the subgroup structure of a one-relator group G to the primitivity rank, a notion introduced by Puder, pi(w) of w. One application of the theory is that every subgroup of G of rank less than pi(w) is free, and another is that when pi(w)>2, G has what we call "negative immersions", which implies that every finitely generated subgroup of G is finitely presented, i.e., coherent, answering a '74 question of Baumslag in this case. The main tools are a nonabelian "rank-nullity" theorem for free groups and some linear programming. This is joint work with Henry Wilton.

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