Doubly degenerate surface groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as follows:


Take a closed surface $\Sigma$ of genus at least $2$ and a pseudo-Anosov map $\varphi:\Sigma\to\Sigma$ and form the mapping torus $$M\_\varphi=\Sigma\times[0,1]/\langle(x,0)\sim(\varphi(x),1)\rangle,$$ which fibers over the circle.
Taking the infinite cyclic cover $N\_\varphi=\Sigma\times\mathbb{R}$ and identifying $\Sigma$ with some lift of the fiber gives an isomorphism $\pi\_1(\Sigma)\cong\pi\_1(N\_\varphi)\subset PSL(2,\mathbb{C})$.
As $N\_\varphi$ has geometrically infinite ends, this is a doubly degenerate surface group.
Since there are lots of closed surfaces and lots of pseudo-Anosov maps, this produces many examples of such a group.


My question is what are some other constructions or standard examples of doubly degenerate surface groups outside of the construction given above. Any references would be extremely helpful.



Example of a doubly degenerate surface group not coming from a pseudo-Anosov mapping torus
Take any pair of measured laminations $\lambda,\mu$ which each fill the surface and are transverse to each other. Take sequences $\sigma\_i,\tau\_i$ in Teichmuller space, such that $\sigma\_i$ converges to $\lambda$ and $\tau\_i$ converges to $\mu$ in compactified Teichmuller space. Thurston's double limit theorem says that the sequence in quasi-fuchsian space parameterized by pair of sequences $\sigma\_i,\tau\_i$ converges to a doubly degenerate surface group with ending laminations $\lambda,\mu$.


