Orbits of curves under the Johnson kernel

Details Orbits of curves under the Johnson kernel Orbits of curves under the Johnson kernel

2011-08-23

arxiv.org/abs/1108.4511v1

Mathematics

Geometric Topology

32 pages, 2 figures

Scientific paper

This paper has two main goals. First, we give a complete, explicit, and computable solution to the problem of when two simple closed curves on a surface are in the same orbit under the Johnson kernel. Second, we develop an approach to the Johnson filtration and the Johnson homomorphism which is functorial under inclusions of subsurfaces. The key point is that the latter reduces the former to a finite computation, which can be carried out by hand. In particular this solves the conjugacy problem in the Johnson kernel for separating twists. One result of independent interest is the complete description of the restriction to subsurfaces of the lower central series of a surface group.

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