In  Masur proved the existence of a dense geodesic in the moduli space for a surface. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove that the set of Perron- Frobenius eigenvectors of positive integer m×m matrices is dense in the positive cone Rm C (these matrices will in fact be the transition matrices of positive automorphisms). We give a proof in the appendix that not every point in the boundary of Outer Space is the limit of a flow line.
Bibliographical noteFunding Information:
2 C. Pfaff was supported first by the ARCHIMEDE Labex (ANR-11-LABX-0033) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir,” managed by the ANR. She was secondly supported by the CRC701 grant of the DFG, supporting the projects B1 and C13 in Bielefeld.
1 The research of Y. Algom-Kfir of was supported by THE ISRAEL SCIENCE FOUNDATION (grant no. 1941/14).
© 2018 European Mathematical Society.
- Outer Space
- Outer automorphism group of the free group
ASJC Scopus subject areas
- Geometry and Topology
- Discrete Mathematics and Combinatorics