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.


1 Answer 1


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 $\lambda,\mu$.


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