Convexity of balls in the outer space

Yulan Qing; Kasra Rafi

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arXiv · 1708.04921

Convexity of balls in the outer space

Yulan Qing·Kasra Rafi

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Abstract

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 $α$, the length of $α$ along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counterexamples.

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Yulan Qing, Kasra Rafi. 2017-08-16. Convexity of balls in the outer space. https://arxiv.org/abs/1708.04921

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