Easy examples of non-arithmetic lattices

Question

I'm starting to look a bit more at discrete subgroups of Lie groups, particularly lattices. A lot is written about arithmetic lattices of Lie groups, and examples abound.

It appears that much less is written about non-arithmetic lattices of Lie groups, and I'm having trouble coming up with even one example in $SL_2(\mathbb{R})$, where I am told that they do exist. Google hasn't been straightforward either.

Are there any easy (simple to define or visualize) examples of non-arithmetic lattices, particularly in $SL_2(\mathbb{R})$, or is there a good resource for such things?

Answer 1

If you have a simple Lie group whose real rank is at least 2, then every lattice in that group is arithmetic. That's a theorem by Margulis. There are known non-arithmetic lattices in some particular families of special unitary groups, constructed by Mostow. You could check out his paper titled Discrete Subgroups of Lie Groups (1985).

Answer 2

Since you are interested in concrete examples of nonarithmetic lattices in $SL(2,R)$, then you should read this paper by Takeuchi: "Arithmetic triangle groups" J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. In this paper Takeuchi gives a complete list of triples $(p,q,r)$, $2\le p\le q\le r\le \infty$ such that the triangle groups defined by the triple are arithmetic (see section 5 of his paper). Anything not on Taakeuchi's list (which is finite and not excessively long), subject to the condition $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} < 1, $$ will define a nonarithmetic lattice in $SL(2,R)$ by the following method: Take a triple $(p,q,r)$ which is not on Takeuchi's list (say, $(2,5,\infty)$ in the noncompact case or $(2, 5, 7)$ in the compact case). Then construct a geodesic triangle in the hyperbolic plane with the angles of the form $\pi/p, \pi/q, \pi/r$. Consider the group $\Gamma^*$ of hyperbolic isometries generated by reflections $s_1, s_2, s_3$ in the sides of this triangle. Then pass to the index 2 orientation preserving subgroup $\Gamma$ of this group (generated by the products $s_1s_2, s_2s_3, s_3 s_1$). This will be an nonarithmetic lattice in $PSL(2,R)$. Now, lift $\Gamma$ to $SL(2,R)$ (the lift could be a nontrivial central extension of $\Gamma$). The result is a nonarithmetic lattice in $SL(2,R)$. You can easily work out a presentation of this lattice using the fact that $\Gamma$ has the presentation $$ < x, y, z| x^p=y^q=z^r=1, xyz=1>, $$ (or just read Takeuchi's paper a bit). These examples are as concrete as they could be. (There are other explicit constructions as well, but they require a tiny bit more work.) It is known that the group $O(p,1)$ contains nonarithmetic lattices for all $p\ge 2$. The construction is due to Gromov and Piatetsky-Shapiro, see here, it is not as explicit but you may still want to read their paper.