The fundamental group $T(p,q)$ of the complement of a $(p,q)$-torus knot (in $S^3$) admits the presentation $\langle a, b \mid a^p=b^q \rangle $. Is $T(p,q)$ linear, i.e., is there a faithful homomorphism of $T(p,q)$ into a matrix group over some field? For example, $T(2,3)$ and $T(3,3)$ are linear, since both can be embedded into the 3-strand braid group $B_3$. (Indeed, $T(2,3)=B_3$.)
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2 Answers
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Torus knot groups are certainly linear. The universal cover of a torus knot complement is Euclidean space, and the action of the group on this cover factors through "translations on euclidean space" cross $PSL_2(\mathbb R)$. Both linear groups.
Generally it's conjectured that all 3-manifold groups are linear. For all I know, there might be a proof already. See What is known about a 3-manifold $M$ when its fundamental group is linear?
answered Oct 24, 2013 at 11:51
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$\begingroup$ Many thanks. I already found now a paper that explains that in detail, namely: "Triangle groups, automorphic forms, and torus knots" by Valdemar V. Tsanov (see section 4.6. arxiv.org/abs/1011.0461 ). $\endgroup$ Oct 24, 2013 at 14:03
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$\begingroup$ Hyperbolic groups were linear... at least as early as Heinz Hopf, I don't think Ian is taking credit for that. :) $\endgroup$ Oct 24, 2013 at 14:32
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5$\begingroup$ Just to avoid confusion, Ryan's use of "hyperbolic groups" is for fundamental groups of manifolds of constant negative curvature, not for "Gromov-hyperbolic groups". $\endgroup$– YCorOct 24, 2013 at 22:52
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By the work of Agol, Liu, Przytycki-Wise and Wise we now know that fundamental groups of irreducible 3-manifolds which are not closed graph manifolds are `virtually special', in particular they are linear over $\Bbb{Z}$.
Oddly enough, the only open case for linearity are fundamental groups of certain closed graph manifolds.
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$\begingroup$ Its situations like this, and the earlier work of Waldhausen and Thurston on "sufficiently large" 3-manifolds that give 3-manifold theory a lovely un-even texture. It's a great study why one should not get too attached to only one tool. $\endgroup$ Oct 25, 2013 at 11:24