getting from [@BO06] as described above, by the $\mathcal{R}ip_{T}(Q)$ for fixed $Q$. Note that $N\rtimes Q$ has property $(T)$.

$\mathcal M^i_m$ is a maximal von Neumann algebra of $\mathcal M$ for every $i$. In particular, when $N\rtimes Q\in \mathcal Rip_{\mathcal T}(Q)$ then for every $i$, $\mathcal M^i_m$ is a non-property (T) maximal von Neumann subalgebra of a property (T) von Neumann algebra $\mathcal M$.

Hyperbolic spaces
=================

Hyperbolic metric spaces
------------------------

I this section we shall mention equivalent definitions for hyperbolic spaces and their relations. In later sections we shall use whichever definition is convenient for the purpose.

Let $\delta\geq 0$. A geodesic triangle in a metric space is said to be $\delta$-slim if each of it’s side is contained in $\delta $ neighborhood of the union of other two sides. A geodesic metric space $X$ is said to be $\delta$-hyperbolic if all geodesic triangles in $X$ are $\delta$ hyperbolic. When a space is said to be hyperbolic it means that it is $\delta$-hyperbolic for some $\delta>0$.

Given a geodesic triangle (formed by joining three points $x,y,z\in X$ by geodesics, denote by $\triangle(x,y,z)$) in a metric space, the triangle equality tells us that there exists unique non-negative numbers $a,b,c$ such 