An extension of the Thurston metric to projective filling currents

2022/10/14 by Sapir, Jenya
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2210.08130

Abstract

We study the geometry of the space of projectivized filling geodesic currents \mathbb P \mathcal C_fill(S). Bonahon showed that Teichmüller space, \mathcal T(S) embeds into \mathbb P \mathcal C_fill(S). We extend the symmetrized Thurston metric from \mathcal T(S) to the entire (projectivized) space of filling currents, and we show that \mathcal T(S) is isometrically embedded into the bigger space. Moreover, we show that there is no quasi-isometric projection back down to \mathcal T(S). Lastly, we study the geometry of a length-minimizing projection from \mathbb P \mathcal C_fill(S) to \mathcal T(S) defined previously by Hensel and the author.

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