# Does moving a small enough distance in Teichmüller space change the marking?

Let $$S_{g}$$ be a genus $$g$$ closed Riemann surface. The Teichmüller space $$\mathcal{T}(S_{g})$$ is the set of all pairs $$(X,\phi)$$ where $$X$$ is a Riemann surface of genus $$g$$ and $$\phi : S_{g} \rightarrow X$$ is a quasiconformal homeorphism, quotient by a certain equivalence relation. We say that two such pairs $$(X_{1},\phi_{1})$$ and $$(X_{2},\phi_{2})$$ are equivalent if there exists a conformal map $$h : X_{1} \rightarrow X_{2}$$ such that $$h \circ \phi_{1}$$ is homotopic to $$\phi_{2}$$. We consider the Teichmüller distance $$d_{T}$$ defined on this space, where $$d_{T} ([(X_{1},\phi_{1})],[(X_{2},\phi_{2})])=\frac{1}{2}logK_{h}$$ such that $$h : X_{1} \rightarrow X_{2}$$ is the unique quasiconformal map in the homotopy class of $$\phi_{2} \circ \phi_{1}^{-1}$$ having the lowest quasiconstant $$K_{h}$$. Let us consider the point $$x=[(S_{g}, id_{S_{g}})] \in \mathcal{T}(S_{g})$$. Does there exist a positive real $$r$$ such that given any point $$y=[(X,\phi)] \in B(x;r)$$ there exists a pair $$(Y,\psi)$$ where $$Y$$ is a Riemann surface with underlying set same as $$S_{g}$$, $$\psi : S_{g} \rightarrow Y$$ is a quasiconformal homeomorphism homotopic to $$id_{S_{g}}$$ and $$(X,\phi)$$ and $$(Y,\psi)$$ are in the same equivalence class, i.e. $$[(X,\phi)]=[(Y,\psi)]$$ ?

Basically, what I am asking is that whether it's true that if we move a small enough distance from a point in Teichmüller space to some other point then the final point has the same making as the starting point.

• This should be true as long as you're in the "thick part" of Teichmüller space.
– HJRW
Sep 10, 2022 at 15:17
• @HJRW can you explain why?
– P.S
Sep 13, 2022 at 6:46
• Here’s what I have in mind. “Changing the marking” means composing with a non-trivial mapping class, which in turn means going round a homotopically non-trivial loop in moduli space. (Here I’m pretending that the map from Teichmüller to moduli space is an honest covering map, instead of an orbifold covering.) So what you ask for in your last paragraph should be equivalent to asking for a lower bound on the injectivity radius of moduli space. This is more or less the definition of the “thick part” of moduli space, and its preimage is the thick part of Teichmüller space.
– HJRW
Sep 13, 2022 at 7:43
• (Warning: this is based entirely on the title of your question and final paragraph. I may easily have missed a subtlety in the body of the question; if so, apologies.)
– HJRW
Sep 13, 2022 at 7:45

Unless I misunderstand your question, you can take $$r = \infty$$. This is because we can choose representatives of marked conformal classes to be "Riemann surface structures" on $$S_g$$. Doing this, the marking maps are all the identity map.

Basically, your equivalence classes are very very large. There is the Riemann surface which is the solutions to $$x^4 + y^4 + z^4 = 0$$ in $$\mathbb{CP}^2$$, and then there is that Riemann surface painted blue, and then there is that Riemann surface crossed with the first strongly inaccessible cardinal, and so on.

To shrink the equivalence classes a bit, we can instead think about Riemann surface structures (maximal atlases of charts) on a fixed surface. Now the marking map is clearer - it is the identity. However, the action of the mapping class group is harder to grasp. Also, our moduli space no longer contains "all" of the Riemann surfaces. Such is life.

• Can you elaborate your definition in the first part of the answer? I suppose the equivalence relation is that two such surfaces are equivalent if there exists a biholomorphic map homotpoic to identity. But I think what I am asking is a little bit different. Consider the standard definition of Teichmuller space using quasiconformal maps as mentioned in the question. I want to know if there is any positive real $r$ such that any point in the ball of radius $r$ centered at $[(S_{g},id_{S_{g}})]$ is of the for $[(Y,\psi)]$ such that $Y$ is Riemann surface with underlying set $S_{g}$ (continued)
– P.S
Sep 12, 2022 at 7:05
• and $\psi$ is homotopic identity. That is I want if $\alpha$ is a hompotopy class of simple closed curves in $S_{g}$ then the corresponding (image of the uniqueTeichmuller map between $S_{g}$ and $Y$ ) homotopy class of curves in $Y$ is also $\alpha$.
– P.S
Sep 12, 2022 at 7:06