Small subgroups of SL\((3,\mathbb Z)\). (English) Zbl 1269.57002

Long, D. D.; Reid, A. W.

Exp. Math. 20, No. 4, 412-425 (2011).

Lubotzky asked the following question: Does every finite-index subgroup of \(SL(n,{\mathbb Z}) (n\geq 3)\) contain two-generator subgroups of finite index? In the paper under review the authors show the following Theorem 1.2 as evidence for an affirmative answer for the case \(n=3.\) Theorem 1.2 states that the group \(SL(3,{\mathbb Z})\) contains a family \(\{N_{j}\}\) of two-generator subgroups of finite index with the property that \(\bigcap N_{j} =1.\) In order to prove this main result, the authors study two one-parameter families of representations of \(\pi_{1}(S^{3} \backslash K)\) into \(SL(n,{\mathbb R})\), where \(K\) is the figure-eight knot. Let \(\Gamma=\pi_{1}(S^{3} \backslash K)\). If we choose generators \(x\) and \(y\) for the fiber group (which is denoted by \(F\)) and \(z\) as the stable letter, then \(\Gamma\) is presented as \[ <x,y,z\mid zxz^{-1}=xy, zyz^{-1}=yxy>. \] Here two representations \(\rho_{k}, \beta_{T}\) are given by explicit images for \(x,y,z\), respectively. The authors prove that, if \(k \in {\mathbb Z}\) (respectively nonzero \(T\in {\mathbb Z}\)) is fixed, then the images of the fiber groups \(\rho_{k}(F)\) (respectively \(\beta_{T}(F)\)) are Zariski-dense subgroups of \(SL(3, {\mathbb R})\). Also they show that for a fixed nonzero integer value of \(T\), the group \(\beta_{T}(F)\) (and therefore \(\beta_{T}(\Gamma)\)) has finite index in \(SL(3,{\mathbb Z})\) and that \(\bigcap _{T>0} \beta_{T}(F)=1.\) After raising a natural question on the existence of an orientable finite-volume hyperbolic 3-manifold \(M\) (or a compact orientable hyperbolizable 3-manifold \(M\) that is not an I-bundle over a surface) for which \(\pi_{1}(M)\) admits a faithful representation into \(SL(3,{\mathbb Z})\), the paper ends with an appendix, which shows the outline of the method for producing the representations \(\rho_{k}, \beta_{T}\) and computations of some indices by using Magma.

Reviewer: Shigeyasu Kamiya (Okayama)

MSC:

57M07	Topological methods in group theory
20H25	Other matrix groups over rings
57M50	General geometric structures on low-dimensional manifolds
20C12	Integral representations of infinite groups

Keywords:

figure-eight knot group; Zariski dense

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