# Generic thinness in finitely generated subgroups of SL_n(Z)

Elena Fuchs and Igor Rivin. IMRN 2017(17), 5385–5414; DOI 10.1093/imrn/rnw136. Online publication 5 August 2016.

The paper asks whether two independently sampled integral determinant-one matrices generate a thin group. Its principal sampling family bounds both the matrix operator norm and its inverse. Ordinary norm balls, symmetrized balls and random walks define different probability models. Separate conclusions concern generic freeness, Zariski density, Hausdorff dimension in dimension two and Lyapunov exponents.

The proposed argument combines Cartan-coordinate volume estimates, lattice-point counting in sectors, exterior-power contraction and ping-pong. Read Section 1 for the sampling model, Section 2 for the two-dimensional case, Sections 3–4 together for the higher-dimensional argument and counting hypotheses, and Section 5 for the separate Lyapunov statement.

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