Isometric Embeddings of Subsets of Boundaries of Teichmüller Spaces of Compact Hyperbolic Riemann Surfaces

Abstract

It is known that every finitely unbranched holomorphic covering π:$$\pi :\tilde{S} \to S$$ of a compact Riemann surface S with genus g ≥ 2 induces an isometric embedding $${{\rm{\Phi}}_{\rm{\pi}}}:Teich\left(S \right) \to Teich\left({\widetilde{S}} \right)$$. By the mutual relations between Strebel rays in Teich(S) and their embeddings in $$Teich\left({\widetilde{S}} \right)$$, we show that the 1st-strata space of the augmented Teichmuller space $$\widehat{Teich}\left(S \right)$$ can be embedded in the augmented Teichmuller space $$\widehat{Teich}\left({\widetilde{S}} \right)$$ isometrically. Furthermore, we show that Φπ induces an isometric embedding from the set Teich(S) B (∞) consisting of Busemann points in the horofunction boundary of Teich(S) into $$Teich{\left({\widetilde{S}} \right)_B}\left(\infty \right)$$ with the detour metric.