For $m,n \gt 0, m\ne n$, $BS(m,n)$ acts on $\mathbb H^2$ by parabolic isometries with a common fixed point. In particular, you can represent it by the action on the upper half plane of

$S = \bigg(\begin{matrix}m/n & 0 \\\ 0 &m/n\end{matrix}\bigg),$ $B = \bigg(\begin{matrix}1 & \alpha \\\ 0 &1\end{matrix}\bigg)$

where $\alpha$ is arbitrary.

There are elements of $BS(m,n)$ which act trivially, but you can lift this to a free action on a topological $T\times \mathbb R$ where $T$ is a $n+m$-regular directed tree with outdegree $m$ and indegree $n$ so that for any path $P \subset T$, $P\times \mathbb R$ is a copy of $\mathbb H^2$ so that the sets $\{p\}\times \mathbb R$ are concentric horocycles (like horizontal lines in the upper half plane).

For $m,n \gt 0, m\ne n$, $BS(m,n)$ acts on $\mathbb H^2$ by isometries with a common ideal fixed point. In particular, you can represent it by the action on the upper half plane of

$S = \bigg(\begin{matrix}\sqrt{m/n} & 0 \\\ 0 &\sqrt{n/m}\end{matrix}\bigg),$ $B = \bigg(\begin{matrix}1 & \alpha \\\ 0 &1\end{matrix}\bigg)$

where $\alpha$ is arbitrary.

There are elements of $BS(m,n)$ which act trivially, but you can lift this to a free action on a topological $T\times \mathbb R$ where $T$ is a $n+m$-regular directed tree with outdegree $m$ and indegree $n$ so that for any path $P \subset T$, $P\times \mathbb R$ is a copy of $\mathbb H^2$ so that the sets $\{p\}\times \mathbb R$ are concentric horocycles (like horizontal lines in the upper half plane).

This is related to the tilings of the hyperbolic plane by horobricks, e.g., there are tiles with arbitrarily small diameter which tile $\mathbb H^2$ which are really fundamental domains of $BS(m,n)$, and analogously there are polyhedra of arbitrary Dehn invariant which tile $\mathbb H^3$.