Fix a compact real surface $S$ of genus $g \geqslant 2$ and look at the space $\mathcal{T}$ of complex structures on $S$ (up to isotopy). I know it has a manifold structure (I think it is called the Teichmuller space?). I wonder if the following holds:
Let $M$ be the set of points in $\mathcal{T}$ such that each $m \in M$ has a trivial automorphism group (as a Riemann surface). Is $M$ open and dense in $\mathcal{T}$?
Does someone know if/why this is true? What can we say about $M$?
Leon Greenberg in
Greenberg, L., Maximal Fuchsian groups, Bull. Am. Math. Soc. 69, 569-573 (1963). ZBL0115.06701.
classified Riemann surfaces $S$ of finite type for which the subset of the Teichmuller space $T(S)$ contains an open and dense subset represented by Riemann surfaces with trivial automorphism group. See Theorems 2 and 3A in his paper. (In fact, he proves more.) Among compact surfaces, genus $g\ge 3$ is necessary and sufficient, hence, the answer to your question. Proofs are very straightforward, one compares dimensions of Teichmuller spaces of certain orbifolds. Greenberg states his results in terms of Fuchsian groups since the orbifold language was unavailable at the time (more precisely, people in the Ahlfors-Bers school were unaware of Satake's work where the concept was introduced in 1956 under the name "V-manifolds"). The key is the following computation:
Suppose that ${\mathcal O}_1$ and ${\mathcal O}_2$ are 2-dimensional oriented orbifolds of finite type such that there exists a finite degree $>1$ orbi-covering map $$ p: {\mathcal O}_2\to {\mathcal O}_1. $$ Then, with a small list of exceptions, $$ dim(T({\mathcal O}_2))> dim (T({\mathcal O}_1)). $$ Here $T$ stands for the Teichmuller space.
For instance, if ${\mathcal O}_2$ is a compact surface of genus $\ge 3$ then there are no exceptions. If ${\mathcal O}_2$ is a compact surface of genus $2$ then there is exactly one exception, corresponding to the case when $p$ is the 2-fold orbi-covering map given by the quotient of ${\mathcal O}_2$ by a hyper-elliptic involution.
Once this is established, the rest of the proof is a straightforward application of the Baire Category Theorem.