What is the implied metric on $SL(2,\mathbb{R})$ in the definition of Fuchsian group?

Question

I'm studying Fuchsian groups, which are by definition discrete subgroup of $SL(2,\mathbb{R})$. Authors often do not tell, though, what topology (metrizable I assume) is taken on $SL(2,\mathbb{R})$ and so the above statement is (possibly) not meaningful.

The metric is supposed to be one such that discreteness would be equivalent to the subgroup acting discontinuously on the hyperbolic plane, as this is a theorem which is commonly used (again, I didn't find any rigorous proof, so I couldn't understand from it what the metric is).

I was wondering - is there a reason why the metric is not mentioned? I get that a lot of the metrics are equivalent, if we think of $SL(2,\mathbb{R})$ as embedded in $\mathbb{R}^4$ and consider any topology defined by a norm (since all norms are equivalent), but $SL(2,\mathbb{R})$ is not a vector space itself. Moreover, one is often interested in invariant metrics, so I couldn't be sure what is the actual intent.

Answer 1

$SL(2,\mathbb R)$ is a matrix group with $4$ coordinates, namely the 4 entries of the matrix. This gives an injection $SL(2,\mathbb R) \mapsto \mathbb R^4$. The topology on $SL(2,\mathbb R)$ is obtained by pulling back the subspace topology on the image of this map (which is actually a submanifold of $\mathbb R^4$).

A subgroup $\Gamma < SL(2,\mathbb R)$ is defined to be discrete if its subspace topology is the discrete topology, meaning the topology in which every point is an open set. Equivalently, $\Gamma$ is discrete if, under the composition of injections $\Gamma \hookrightarrow SL(2,\mathbb R) \hookrightarrow \mathbb R^4$, the image is a discrete subset of $\mathbb R^4$.

All of this has been done without any metric. Nonetheless, $SL(2,\mathbb R)$ is a Lie group and there is a standard way to put metrics, and a metric topology, on a Lie group. Here are those methods:

1. To put a left-invariant Riemannian metric on a Lie group, pick a positive definite quadratic form on the tangent space of the identity element, and then push it forward to every other point by the derivative of the left multiplication map.
2. To put a metric-space structure on a (connected) Riemannian manifold, take the infimum of path lengths between two points.
3. To put a topology on a metric space, use open balls as a basis.

And although there was a choice in (1), the topology that you get in the end is independent of the choice. Furthermore, for a Lie group like $SL(2,\mathbb R)$ which is explicitly presented as a matrix group, the topology just described is the same as the topology originally described using matrix coordinates. So, discreteness of subgroups is well-defined independent of all choices.