Groups, Geometry, and Dynamics

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Published online first: 2021-07-30
DOI: 10.4171/GGD/615

Convexity of balls in outer space

Yulan Qing[1] and Kasra Rafi[2]

(1) Fudan University, Shanghai, China
(2) University of Toronto, Canada

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $\alpha$, the length of $\alpha$ along a balanced folding path is not larger than the maximum of its lengths at the endpoints. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counter examples.

Keywords: Outer space, Out($\mathbb{F}_n$), folding path

Qing Yulan, Rafi Kasra: Convexity of balls in outer space. Groups Geom. Dyn. Electronically published on July 30, 2021. doi: 10.4171/GGD/615 (to appear in print)