I am interested in groups of isometries which act discretely on uniform spaces of constant curvature, ie $\mathbb{S}^n$, $\mathbb{E}^n$, and $\mathbb{H}^n$. I believe I am right that any such group acting on $\mathbb{S}^n$ or $\mathbb{E}^n$ can be realised as a finite index subgroup of a discrete group generated by reflections (a finite or affine Coxeter group respectively).
Question: can any discrete group $G$ of isometries acting on $\mathbb{H}^n$ be realised as a finite index subgroup of a Coxeter group acting on $\mathbb{H}^n$?
On the one hand, any isometry of $\mathbb{H}^n$ can be decomposed as a sequence of reflections, so intuitively one could take a set of generators for $G$ and replace any which are not reflections be a set of suitable reflections, but a priori it is not clear that the resulting group will be discrete. On the other hand we know from the classification of irreducible hyperbolic Coxeter groups, that they only exist in dimensions $3,\dots,10$ which suggests there may be discrete groups of isometries acting essentially on $\mathbb{H}^n$ for $n>10$ which aren't subgroups of Coxeter groups.
An explicit answer would be great, but alternatively a specific reference where I can find the answer would be useful. I'm pretty sure this isn't answered in Humphreys, Bourbaki, or Brown for example.
Best Answer
Your claim that every discrete group of isometries of $\mathbb E^2$ is equal to a finite index sugroup of a Coxeter group is not true. One can infer this from a dimension count, as follows.
Let's consider the space of discrete groups $\Gamma$ of isometries of $\mathbb E^2$ with the following properties:
This space is 2-dimensional, as you can see by noticing that there exists an element $T_{a,b}$ in the group (meaning that $T_{(a,b)}(x,y)=(x+a,y+b)$) such that $(a,b)$ is contained in the region defined by the inequalities $a^2+b^2 \ge 1$ and $0 \le a \le 1$, and furthermore this element is almost unique: $\Gamma$ contains at most 2 such elements $T_{(a,b)}$, and 2 of them occur only if $(a,b)$ lies on the boundary of the region, meaning that $a^2+b^2=1$ or $a=0$ or $a=1$.
The Coxeter triangle groups are all rigid up to normalization, and so their finite index subgroups form a countable subset of the space of discrete groups.
The only Coxeter groups that are not triangle groups are the rectangle reflection groups, and they form just a 1-dimensional space. Their finite index subgroups therefore form a 1-dimensional subset of the space of discrete groups, namely those for which there exists some element of the form $T_{(0,b)|$, $b \ge 1$. Your up/down, left/right example in your comment does lie in this subspace.
Using this analysis, for an explicit example that is not a finite index subgroup of a Coxeter group, take the group generated by $T_{(1,0)}$, $T_{(1/2,1)}$.
A similar dimension count applies to discrete groups of isometries of $\mathbb H^2$. This gets into the topic of hyperbolic surfaces and hyperbolic 2-orbifolds and their Teichmuller spaces. I'll give just the simplest example (which is already too complicated to describe in full detail, so I'll have to leave a lot of claims unproved). Consider the abstract group $\pi_1(S_2)$ which is the fundamental group of the closed oriented surface $S_2$ of genus $2$, defined by the presentation $$\pi_1(S_2) = \langle a,b,c,d \mid aba^{-1}b^{-1}cdc^{-1}d^{-1} \rangle $$ This group has a Teichmuller space, which is the space of (suitably normalized) discrete subgroups of isometries of $\mathbb H^2$ that are isomorphic to $\pi_1(S_2)$. One can do a dimension count to show that this space is 6-dimensional. Also, the Euler characteristic of $S_g$ is equal to $\chi(S_g)=2-2g$ in general, and $\chi(S_2)=-2$.
If $\pi_1(S_2)$ is isomorphic to a finite index subgroup of a Coxeter group, then the corresponding reflection polygon $P$ has a rational Euler characteristic $\chi(P)$ which is a rational number strictly between $-2$ and $0$. These reflection polygons can be explicitly enumerated, and the dimensions of their Teichmuller spaces all calculated explicitly, and one sees that in each case the dimension is strictly less than $6$.