Since my previous question

Hyperelliptic loci in Teichmueller spaces

resulted in two quick and helpful replies, let me ask another question in a similar vein:

A smooth compact complex curve is called trigonal, if it is a triple cover of the projective line. Let ${\mathcal X}_g$ be the trigonal locus in the moduli space of smooth genus $g$ curves (not sure what the standard notation is). What can be said about the topology of the preimage of ${\mathcal X}_g$ in the Teichmueller space? In particular, is it connected? If not, is there a description of its connected components in terms of mapping class groups similar to the one in the hyperelliptic case? Can one say anything reasonable about the fundamental groups of the connected components?

Since my previous question

Hyperelliptic loci in Teichmueller spaces

resulted in two quick and helpful replies, let me ask another question in a similar vein:

A smooth compact complex curve is called trigonal, if it is a triple cover of the projective line. Let ${\mathcal X}_g$ be the trigonal locus in the moduli space of smooth genus $g$ curves (not sure what the standard notation is). What can be said about the topology of the preimage of ${\mathcal X}_g$ in the Teichmueller space? In particular, is it connected? If not, is there a description of its connected components in terms of mapping class groups similar to the one in the hyperelliptic case? Can one say anything reasonable about the fundamental groups of the connected components?