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On the self-intersections of curves deep in the lower central series of a surface group

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Abstract

We give various estimates of the minimal number of self-intersections of a nontrivial element of the kth term of the lower central series and derived series of the fundamental group of a surface. As an application, we obtain a new topological proof of the fact that free groups and fundamental groups of closed surfaces are residually nilpotent. Along the way, we prove that a nontrivial element of the kth term of the lower central series of a nonabelian free group has to have word length at least k in a free generating set.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL, 60637-1514, USA

    Justin Malestein

  2. Department of Mathematics, MIT, 2-306, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Andrew Putman

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  1. Justin Malestein
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  2. Andrew Putman
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Correspondence to Andrew Putman.

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Malestein, J., Putman, A. On the self-intersections of curves deep in the lower central series of a surface group. Geom Dedicata 149, 73–84 (2010). https://doi.org/10.1007/s10711-010-9465-z

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  • DOI: https://doi.org/10.1007/s10711-010-9465-z

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