# Example of a doubly degenerate surface group not coming from a pseudo-Anosov mapping torus

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.

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 $\lambda,\mu$.