The extrinsic primitive torsion problem

Bou-Rabee, Khalid; Hooper, W Patrick

doi:10.2140/agt.2020.20.3329

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The extrinsic primitive torsion problem

Khalid Bou-Rabee and W Patrick Hooper

Algebraic & Geometric Topology 20 (2020) 3329–3376

DOI: 10.2140/agt.2020.20.3329

Abstract

Let $P_{k}$ be the subgroup generated by $k^{th}$ powers of primitive elements in $F_{r}$ , the free group of rank $r$ . We show that $F_{2} ∕ P_{k}$ is finite if and only if $k$ is $1$ , $2$ or $3$ . We also fully characterize $F_{2} ∕ P_{k}$ for $k = 2, 3, 4$ . In particular, we give a faithful $9$ –dimensional representation of $F_{2} ∕ P_{4}$ with infinite image.

Keywords

Burnside problem, primitive elements, characteristic subgroups, square-tiled surface

Mathematical Subject Classification 2010

Primary: 20F05, 20F65

Secondary: 20F38

References

Publication

Received: 11 December 2018

Revised: 30 October 2019

Accepted: 10 January 2020

Published: 29 December 2020

Authors

Khalid Bou-Rabee
Department of Mathematics The City College of New York New York, NY United States	Department of Mathematics The Graduate Center, CUNY New York, NY United States
https://sites.google.com/site/khalidmath/
W Patrick Hooper
Department of Mathematics The City College of New York New York, NY United States	Department of Mathematics The Graduate Center, CUNY New York, NY United States
http://wphooper.com/

Volume 20, issue 7 (2020)