Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces. (English) Zbl 1166.57012

Penner, Robert (ed.) et al., Groups of diffeomorphisms in honor of Shigeyuki Morita on the occasion of his 60th birthday. Based on the international symposium on groups and diffeomorphisms 2006, Tokyo, Japan, September 11–15, 2006. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-48-8/hbk). Advanced Studies in Pure Mathematics 52, 443-468 (2008).
In an earlier paper [Duke Math. J. 70, No. 3, 699–726 (1993; Zbl 0801.57011)], the author defined a family of maps called traces. They are defined on certain modules of derivations of a Lie algebra associated with the mapping class group of a surface \(S\) with one boundary component and have values in a symmetric power of the first cohomology of the surface \(S\).
The main result of the present paper is to define a group version of the above traces. They are defined on a subgroup of the automorphism group of the quotients of the (free) fundamental group of \(S\) by the lower central series and have values in a symmetric power of the first rational cohomology of \(S\). These homomorphism extend to cocycles (or crossed homomorphisms) defined on the whole automorphism group mentioned above.
Next the author applies the above results to the study of the group \(\mathcal H_{g,1}\) of homology cobordism classes of homology cylinders of \(S\) defined by S. Garoufalidis and J. Levine [Proceedings of Symposia in Pure Mathematics 73, 173–203 (2005; Zbl 1086.57013)]. For example, he defines cohomology classes of \(\mathcal H_{g,1}\) using the group version of the traces.
This very interesting paper contains several open problems and conjectures.
For the entire collection see [Zbl 1154.53004].


57M99 General low-dimensional topology
53D30 Symplectic structures of moduli spaces