The Tits Cone - Geometric Understanding I know that the definition of the Tits cone is $Y=\bigcup_{w\in W}{wC}$ with W the Coxeter Groups and C the fundamental chamber. One theorem says that Y is the whole space if W is finite. But how can i understand this geometrically and how can i see/understand that if W is not finite that then Y is not the whole space (instead of the technically proofs).
Thank you for understanding.
 A: First of all, it should be $Y = \bigcup_{w \in W} w\overline{C}$, and not $Y = \bigcup_{w \in W} wC$.
To prove that $Y$ equals the whole space $V$, one can proceed as follows. Consider the complement $X$ of the union of all the hyperplanes
$$H = \{ x : wsw^{-1}x = x\},\quad w \in W, s \in S$$
and call the path-connected components of this set chambers. Assuming that $X$ is dense in $V$ (trivially true if $W$ is finite), it suffices to prove that any chamber $C'$ is of the form $C' = wC$ for some $w \in W$. 
Let $d(C,C')$ equal the number of hyperplanes separating $C,C'$, and show that $C' = wC$ for some $w \in W$ by induction over $d(C,C')$ (show that there exists a gallery $(C = C_0,C_1,\ldots,C_n = C')$ of length $n = d(C,C')$ connecting $C,C'$, and prove that $C_1 = sC$ for some $s \in S$). This assumes that $d(C,C') < \infty$, which fails for general Coxeter groups.
To see how the assumptions of the above proof can fail, consider the example $W = \text{PGL}_2(\mathbb{Z})$. (see the book $Buildings$ by Brown) Here $S = \{s_1,s_2,s_3\}$ with
$$ s_1 = \begin{bmatrix} & 1 \\ 1& \end{bmatrix},\quad s_2 = \begin{bmatrix} -1 & 1 \\ & 1 \end{bmatrix},\quad s_3 = \begin{bmatrix} -1 & \\ & 1\end{bmatrix}$$
Tits representation for $(W,S)$ can be identified with the space of binary real quadratic forms, or equivalently, with the space $V \subseteq \text{M}_2(\mathbb{R})$ of real symmetric $2x2$ matrices, where the action of $w \in \text{PGL}_2(\mathbb{Z})$ on a matrix $A \in V$ is given by
$$ wA = \widehat{w}A\widehat{w}^t$$
where $\widehat{w} \in \text{GL}_2(\mathbb{Z})$ is any lift of $w$. Under this identification, the canonical basis vectors $e_i := e_{s_i}$ are given by the matrices
$$ e_1 = \begin{pmatrix} 1 & \\ & -1\end{pmatrix},\quad e_2 = \begin{pmatrix} -1 & -1 \\ -1 & \end{pmatrix},\quad e_3 = \begin{pmatrix}  & 1 \\ 1 &\end{pmatrix}$$
The fundamental chamber is given by
$$ C = \{ x \in V : \forall i = 1,2,3\ B(e_i,x) > 0 \} = \{ \begin{pmatrix}a & b \\ b & c\end{pmatrix} : a > c > 2b > 0\} $$
where $B$ is inner product on $V$ defined by $B(e_s,e_t) = -\cos \frac{\pi}{m(s,t)}$ and given explicitly by
$$ B\left(\begin{pmatrix} a & b \\ b & c\end{pmatrix},\begin{pmatrix} a' & b' \\ b' & c'\end{pmatrix}\right) = bb' - \frac{1}{2}(ad'+a'd)$$
It's easy to see that
$$ C \subseteq P := \{ A \in V : \text{det}(A) > 0, \text{tr}(A) > 0 \}$$
and that $P$ is precisely the subset of positive definite matrices (or forms). Since the action of $W$ preserves the signature, it follows that the translates $wC$ are also all contained in $P$. In fact, $Y = \overline{P}$ is the cone of semi-positive definite matrices (resp. forms).
Moreover, the positive-definite matrix
$$A = \begin{pmatrix} 1 & \\ & 1\end{pmatrix}$$
and the negative-definite matrix $-A$ are separated by the infinitely many hyperplanes
$$H_n = \{ x \in V : B(\begin{bmatrix} 1 & n \\ & 1\end{bmatrix}e_1, x) = 0 \} = \{ \begin{pmatrix} a & b \\ b & c\end{pmatrix} : nb - \frac{1}{2}((n^2-1)c + a) = 0\},\quad n \in \mathbb{N}$$
For a more elementary example, consider $W = D_\infty$ the infinite dihedral group.
