Adding high powered relations to large groups

2005/12/15 by Marc Lackenby, Lackenby, Marc
Mathematics · #20F65 #57M07 #57N10 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F65 #msc:57M07 #msc:57N10

paper · pdf · doi:10.48550/arxiv.math/0512356

15 pages, 7 figures; v2: minor corrections and improved exposition; to appear in Mathematical Research Letters

arxiv created 2007/03/29 · arxiv updated 2009/12/01

Abstract

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. The main theorem of the paper is as follows. Let G be a finitely generated, large group and let g₁,...,g_r be a collection of elements of G. Then G/<<g₁ⁿ,...,g_rⁿ>> is also large, for infinitely many integers n. Furthermore, when G is free, this holds for all but finitely many n. These results have the following application to Dehn surgery. Let M be a compact orientable 3-manifold with boundary a torus. Suppose that the 3-manifold obtained by Dehn filling some slope on the boundary has large fundamental group. Then this is true for infinitely many filling slopes.

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