  In the last years, there has been a large amount of research on embeddability
properties of finitely generated hyperbolic groups. In this paper, we elaborate
on the more general class of locally compact hyperbolic groups. We compute the
equivariant $L_p$-compression in a number of locally compact examples, such as
the groups $SO(n,1)$: by proving that the equivariant $L_p$-compression of a
locally compact compactly generated group is minimal for $p=2$, we calculate
all equivariant $L_p$-compressions of $SO(n,1)$. Next, we show that although
there are locally compact, non-discrete hyperbolic groups $G$ with Kazhdan's
property ($T$), it is true that any locally compact hyperbolic group admits a
proper affine isometric action on an $L_p$-space for $p$ larger than the
Ahlfors regular conformal dimension of $\partial G$. This answers a question
asked by Yves de Cornulier. Finally, we elaborate on the locally compact
version of property $(A)$ and show that, as in the discrete case, a locally
compact second countable group has property (A) if its non-equivariant
compression is greater than 1/2.
