Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any elements $x_1,...x_n$ and all real numbers $c_1,...,c_n$ c_1,...,c_n$, with $c_1+...+c_n=0$, the following inequality holds: $$ \sum_{i=1}^{n}\sum_{j=1}^{n} c_ic_j\psi(x_i,x_j)\leq 0. $$
Let $G$ be a discrete group. Recall that a function $G\rightarrow \mathbb{R}$ is conditionally of negative type if the kernel $\psi$, defined by $\psi(g,h)= \psi(h^{−1}g)$ is conditionaly of negative type.
Does there exist class of discrete groups which admit an explicit description of functions which are conditionaly of negative type?
