The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a\_s,b\_s)\_{s \in S}$ one can attach a Hecke algebra $\mathcal{H}(W,S)$ (this is an exercise in Bourbaki's Groupes et algèbres de Lie IV-VI) with generators $T\_w$ ($w \in W$) and relations


$$ \begin{align\*} T\_w T\_{w'} & = T\_{ww'} \text{ if } \ell(w)+\ell(w') = \ell(ww') \\ T\_s^2 & = a\_s + b\_s T\_s \end{align\*} $$


Sometimes, these abstract Hecke algebras turn out to be isomorphic to `convolution type Hecke algebras'


$$ \mathcal{H}(G,B) = \text{End}\_G(\text{ind}^G\_B \Bbb{1}) $$


for a group $G$ and a subgroup $B$ (I am being deliberately vague here), and one therefore gets a functor
$$ \text{Rep}(G) \longrightarrow \text{R-Mod}(\mathcal{H}(W,S)),\quad V \mapsto V^B $$
relating representations of the abstract Hecke algebra and of the group $G$. This is for instance the case if $(W,S)$ comes from a BN-pair (by another Bourbaki exercise).



Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?
$PGL\_2(\mathbf{Z})$ is a Coxeter group with presentation $\langle s\_1,s\_2,s\_3 \mid s\_i^2=(s\_1s\_2)^3=(s\_1s\_3)^2=1 \rangle$, i.e. with labels $\{2,3,\infty\}$. To each Coxeter group $W$ with labels in $\{2,3,4,6,\infty\}$ there exists a Kac--Moody $G$ having a (twin) BN-pair with Weyl Group $W$, as constructed by J. Tits. In particular, $PGL\_2(\mathbf{Z})$ is the Weyl group of a Kac-Moody group of rank 3. 


While the definition of a Kac-Moody group might appear quite technical, there are two ways of handwaving to convey a flavour of what a Kac-Moody group is like:


(i) A semisimple complex Lie algebra has a nice presentation which can be written down from its Dynkin diagram. Using this presentation for more general Dynkin diagrams, one gets an infinite-dimensional complex Lie algebra $L$ generated by some generators $\langle e\_i, f\_i, h\_i\rangle$. Then the associated adjoint complex Kac-Moody group is by definition the subgroup of Aut$(L)$ generated by ad$(t\cdot e\_i)$, ad $(t\cdot f\_i)$ for $t \in \mathbf C$. 


(ii) One can write down a presentation for $SL\_n(\mathbf C)$ with generators the elementary matrices and certain relations which can be read off from the group's associated Dynkin diagram. Then a Kac-Moody group can similiarly be written down in terms of generators and relations for more general Dynkin diagrams. 


Caprace-Rémy's "[Groups with a twin root datum](http://bremy.perso.math.cnrs.fr/RGD.pdf)" gives a very nice introduction to these matters. 


On the other hand, by Tits' famous classification of spherical buildings there do not exist thick buildings of types $H\_3$ or $H\_4$, hence no $BN$-pairs of this type, see e.g. this [paper](http://www1.mat.uniroma1.it/~combinat/quaderni/quad12E/12E_weiss.pdf) by R. Weiss. 


