Asymmetry of Outer space

Yael Algom-Kfir; Mladen Bestvina

doi:10.1007/s10711-011-9591-2

Asymmetry of Outer space

Yael Algom-Kfir
, Mladen Bestvina

Research output: Contribution to journal › Article › peer-review

Abstract

We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm {double pipe}·{double pipe}^L that induces d. There is an Out(F_n)-invariant "potential" Ψ defined on Outer space such that when {double pipe}·{double pipe}^L is corrected by dΨ, the resulting norm is quasi-symmetric. As an application, we give new proofs of two theorems of Handel-Mosher, that d is quasi-symmetric when restricted to a thick part of Outer space, and that there is a uniform bound, depending only on the rank, on the ratio of logs of growth rates of any irreducible f ε Out(F_n) and its inverse.

Original language	English
Pages (from-to)	81-92
Number of pages	12
Journal	Geometriae Dedicata
Volume	156
Issue number	1
DOIs	https://doi.org/10.1007/s10711-011-9591-2
State	Published - Feb 2012
Externally published	Yes

Bibliographical note

Funding Information:
Acknowledgments We thank Bert Wiest and the referee for helpful comments. The second author gratefully acknowledges the support by the National Science Foundation

Keywords

Automorphisms of the free group
Outer space

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s10711-011-9591-2

Cite this

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AU - Algom-Kfir, Yael

AU - Bestvina, Mladen

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N2 - We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm {double pipe}·{double pipe}L that induces d. There is an Out(Fn)-invariant "potential" Ψ defined on Outer space such that when {double pipe}·{double pipe}L is corrected by dΨ, the resulting norm is quasi-symmetric. As an application, we give new proofs of two theorems of Handel-Mosher, that d is quasi-symmetric when restricted to a thick part of Outer space, and that there is a uniform bound, depending only on the rank, on the ratio of logs of growth rates of any irreducible f ε Out(Fn) and its inverse.

AB - We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm {double pipe}·{double pipe}L that induces d. There is an Out(Fn)-invariant "potential" Ψ defined on Outer space such that when {double pipe}·{double pipe}L is corrected by dΨ, the resulting norm is quasi-symmetric. As an application, we give new proofs of two theorems of Handel-Mosher, that d is quasi-symmetric when restricted to a thick part of Outer space, and that there is a uniform bound, depending only on the rank, on the ratio of logs of growth rates of any irreducible f ε Out(Fn) and its inverse.

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Asymmetry of Outer space

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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