I'm reading the article by Geoffrey Mess The Torelli groups for genus 2 and 3 surfaces (pp. 785 - 786), and I'm trying to understand the part that concerns genus 3. We've got a map $UT(S) \to T_3/\mathcal{I}_3$, where $UT(S)$ is a unit tangent bundle over a curve of genus 2, $T_3$ is Teichmuller space and $\mathcal{I}_3$ is Torelli group, this map induces the following one $\pi_1(UT(S)) \to \pi_1(T_3/\mathcal{I}_3) \cong \mathcal{I}_3$. It's interesting to understand the images of the generators of $\pi_1(UT(S))$ in $\mathcal{I}_3$. I'm trying to do that by studying deck transformations of Teichmuller space and understanding which elements of Torreli group acts the same.
$UT(S) \to T_3/\mathcal{I}_3$ is built in the following way. We just take a point in Siegel space $\mathcal{Z}_3$ that corresponds to a Jacobian of $C \cup E$, where $C$ is a genus 2 curve and $E$ is a torus, then we can move a point where these two curves intersect over the whole genus 2 curve not changing the Jacobian, and then by adding annulli instead of those points we get genus three surfaces that can be considered as points in Torelli space.
Using Fenchel–Nielsen coordinates I guess that I can show that the generator $s$ that corresponds to going around the fiber (that is a circle) maps to Dehn twist of a separating curve. However, I can't understand how to find the images of other generators. When I try to study the process of dragging the torus around some loop corresponding to some (surface) generator, I get a problem with understanding how the coordinates change and what path we are going in Teichmuller space, so it seems that this is not the right way to look at this problem.
