Drutu, Sapir, Osin showed that
a finitely generated group $G$ is strongly hyperbolic relative to a finite collection $\mathcal{H}$ of subgroups if and only if any asymptotic cone is tree-graded with respect to the collection of pieces given by ultralimits of elements in $\mathcal{L}\mathcal{H}$, where $\mathcal{L}\mathcal{H}$ is the collection of all left cosets of $H\_i\in\mathcal{H}$.


In particular, if $G$ is SRH, then any asymptotic cone of $G$ has global cut-points


Furthermore, Behrstock, Drutu, Mosher talked about thickness of groups (in particular, these groups are not strongly relatively hyperbolic). In their paper, they mentioned that mapping class groups are thick, weakly relatively hyperbolic and their asymptotic cones have global cut-points.



> 
> **Question.** Is there any relation between groups which are weakly relatively hyperbolic and their asymptotic cones? For example, if $G$
> is WRH, then does any asymptotic cone have cut points?
> 
> 
> 



Weakly relatively hyperbolicity and asymptotic cone
A finitely generated group is always weakly hyperbolic relative to itself. But here the hyperbolic space you get is just a point... A more interesting example: a free abelian group $\mathbb{Z}^n$ is weakly hyperbolic relative to $\mathbb{Z}^{n-1}$. But here the hyperbolic space you get is a line, so it still may be considered as an elementary example. More generally, in the direct sum of finitely many finitely generated groups, if one of them is weakly relatively hyperbolic, then so is the direct sum. For instance, $\mathbb{F}\_2 \times \mathbb{Z}^2$ is weakly hyperbolic relative to $\mathbb{Z}^2$ and the hyperbolic space you get is a tree, so it is a non-elementary example. Clearly, its asymptotic cones have no cut points.


Observe that, if $X$ is the asymptotic cone of a group, then $X \times \mathbb{R}^n$ and $X \times T$ (where $T$ is a universal real tree) are asymptotic cones of weakly relatively hyperbolic groups. It seems unlikely to exhibit a property satisfied specifically by asymptotic cones of weakly relatively hyperbolic groups.


