November 2022 Non-freeness of Groups Generated by Two Parabolic Elements with Small Rational Parameters
Sang-hyun Kim, Thomas Koberda
Michigan Math. J. 71(4): 809-833 (November 2022). DOI: 10.1307/mmj/20205868

Abstract

Let qC, let

a=1011,bq=1q01,

and let Gq<SL2(C) be the group generated by a and bq. In this paper, we study the problem of determining when the group Gq is not free for |q|<4 rational. We give a robust computational criterion, which allows us to prove that if q=s/r for |s|27, then Gq is non-free with the possible exception of s=24. In this latter case, we prove that the set of denominators rN for which G24/r is non-free has natural density 1. For a general numerator s>27, we prove that the lower density of denominators rN for which Gs/r is non-free has a lower bound

1(111s)n=1(14s2n1).

Finally, we show that for a fixed s, there are arbitrarily long sequences of consecutive denominators r such that Gs/r is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.

Citation

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Sang-hyun Kim. Thomas Koberda. "Non-freeness of Groups Generated by Two Parabolic Elements with Small Rational Parameters." Michigan Math. J. 71 (4) 809 - 833, November 2022. https://doi.org/10.1307/mmj/20205868

Information

Received: 3 February 2020; Revised: 14 April 2020; Published: November 2022
First available in Project Euclid: 10 August 2021

MathSciNet: MR4505367
zbMATH: 07624523
Digital Object Identifier: 10.1307/mmj/20205868

Subjects:
Primary: 30F35 , 30F40
Secondary: 11J70 , 20E05

Rights: Copyright © 2022 The University of Michigan

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Vol.71 • No. 4 • November 2022
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