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

The universal commensurability Busemann horofunction compactified Teichmüller space

“…It is proved in [11] that a covering α can induce an isometric embedding from $$\partial ^{B}_{\text{hor}}\mathcal {T}(S)$$ to $$ \partial ^{B}_{\text{hor}}\mathcal {T}(\widetilde{S}_{g(\alpha )})$$ with the detour metric. Furthermore, we can obtain the following theorem.…”

“…A ray $$r_{\infty }$$ in $$\mathcal {T}_{\infty }(S)$$ is defined as the equivalent class of rays in the stratifications $$\mathcal {T}(\widetilde{S}_{g(\alpha )})$$ for $$\alpha \in \text{Mor}(\mathcal {C}(S))$$. Since it is proved in [11] that a quadratic differential $$q\ne 0$$ on S is Jenkins–Strebel if and only if its lifting differential $$\widetilde{q}$$ on $$\widetilde{S}_{g(\alpha )}$$ is Jenkins–Strebel, then the image of a Jenkins–Strebel ray r in $$\mathcal {T}(S)$$ under the isometric embedding $$\Gamma _{\alpha }:\mathcal {T}(S)\rightarrow \mathcal {T}(\widetilde{S}_{g(\alpha )})$$ is still Jenkins–Strebel in $$\mathcal {T}(\widetilde{S}_{g(\alpha )})$$. We call a ray in $$\mathcal {T}_{\infty }(S)$$ Jenkins–Strebel if it is the equivalent class of the Jenkins—Strebel rays in the stratifications …”

“…The following lemma [11] plays an important role in the definition of $$\mathcal {T}^{B}_{\infty }(S)$$. Lemma Let $$\alpha :\widetilde{S}_{g(\alpha )}\rightarrow S$$ be an unbranched finite covering of S .…”

“…Then, we consider the action of the universal commensurability modular group Mod ∞ (𝑆) on  𝐵 ∞ (𝑆). The following lemma [11] plays an important role in the definition of  𝐵 ∞ (𝑆).…”

“…It is proved in [11] that a covering 𝛼 can induce an isometric embedding from 𝜕 𝐵 hor  (𝑆) to 𝜕 𝐵 hor  ( S𝑔(𝛼) ) with the detour metric. Furthermore, we can obtain the following theorem.…”