Quantifying the Residual Properties of Gamma-Limit Groups

Details Quantifying the Residual Properties of Gamma-Limit Groups Quantifying the Residual Properties of Gamma-Limit Groups

2011-10-14

arxiv.org/abs/1110.3286v1

Mathematics

Group Theory

22 pages, 6 figures

Scientific paper

Let Gamma be a fixed hyperbolic group. The Gamma-limit groups of Sela are exactly the finitely generated, fully residually Gamma groups. We give a new invariant of Gamma-limit groups called Gamma-discriminating complexity and show that the Gamma-discriminating complexity of any Gamma-limit group is asymptotically dominated by a polynomial. Our proof relies on an embedding theorem of Kharlampovich-Myasnikov which states that a Gamma-limit group embeds in an iterated extension of centralizers over Gamma. The result then follows from our proof that if G is an iterated extension of centralizers over Gamma, the G-discriminating complexity of a rank n extension of a cyclic centralizer of G is asymptotically dominated by a polynomial of degree n.

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