#### Volume 20, issue 7 (2020)

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The extrinsic primitive torsion problem

### Khalid Bou-Rabee and W Patrick Hooper

Algebraic & Geometric Topology 20 (2020) 3329–3376
##### Abstract

Let ${P}_{k}$ be the subgroup generated by 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