Quantifying the Residual Properties of Gamma-Limit Groups

TL;DR

This paper introduces a new invariant called Gamma-discriminating complexity for Gamma-limit groups, showing it is asymptotically polynomial, which advances understanding of their residual properties.

Contribution

It defines Gamma-discriminating complexity and proves it is asymptotically dominated by a polynomial for Gamma-limit groups, using embedding theorems and extensions of centralizers.

Findings

Gamma-discriminating complexity is asymptotically polynomial.

Gamma-limit groups embed in iterated extensions of centralizers.

The degree of polynomial bound depends on the rank of the extension.

Abstract

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.