The universal commensurability Busemann horofunction compactified Teichmüller space

In this paper, the direct limit T∞B(S)$\mathcal {T}_{\infty }^{B}(S)$ of Busemann horofunction compactified Teichmülller spaces is introduced. It is shown that the action of the universal commensurability modular group Mod∞(S)$\text{Mod}_{\infty }(S)$ on T∞B(S)$\mathcal {T}_{\infty }^{B}(S)$ is isometric and for any point in the universal commensurability Teichmüller space T∞(S)$\mathcal {T}_{\infty }(S)$, its orbit under this action is dense in T∞B(S)$\mathcal {T}^{B}_{\infty }(S)$. Furthermore, we construct the direct limit M∞B(S)$\mathcal {M}^{B}_{\infty }(S)$ of Busemann horofunction compactified moduli spaces by the characteristic towers and show that the subgroup Caut(π1(S))$\text{Caut}(\pi _{1}(S))$ of Mod∞(S)$\text{Mod}_{\infty }(S)$ acts on T∞B(S)$\mathcal {T}^{B}_{\infty }(S)$ to produce M∞B(S)$\mathcal {M}^{B}_{\infty }(S)$ as the quotient. Finally, we get a formula of the limit distance between two Jenkins–Strebel rays in the universal commensurability Teichmüller space.