823 research outputs found

    Index realization for automorphisms of free groups

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    For any surface Σ\Sigma of genus g1g \geq 1 and (essentially) any collection of positive integers i1,i2,,ii_1, i_2, \ldots, i_\ell with i1++i=4g4i_1+\cdots +i_\ell = 4g-4 Masur and Smillie have shown that there exists a pseudo-Anosov homeomorphism h:ΣΣh:\Sigma \to \Sigma with precisely \ell singularities S1,,SS_1, \ldots, S_\ell in its stable foliation L\cal L, such that L\cal L has precisely ik+2i_k+2 separatrices raying out from each SkS_k. In this paper we prove the analogue of this result for automorphisms of a free group FNF_N, where "pseudo-Anosov homeomorphism" is replaced by "fully irreducible automorphism" and the Gauss-Bonnet equality i1++i=4g4i_1+\cdots +i_\ell = 4g-4 is replaced by the index inequality i1++i2N2i_1+\cdots +i_\ell \leq 2N-2 from Gaboriau, Jaeger, Levitt and Lustig.Comment: 19 pages, 3 figures, 1 table, revised version for publicatio

    Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space

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    We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.Comment: some minor updates and revisions; 18 pages, no figures; to appear in the Pacific Journal of Mathematic

    High distance Heegaard splittings via fat train tracks

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    We define "fat" train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in the three sphere.Comment: 25 pages no figures. to appear in Proceedings of "Knots Groups and 3-manifolds" Marseilles France 200

    Perron-Frobenius theory and frequency convergence for reducible substitutions

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    We prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory.Comment: v.2 Minor revisions for submissio
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