We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of R-trees. This gives rise to a characterisation of closed connected subalgebras of finite products of complete R-trees up to bilipschitz equivalence. Spaces of this sort arise as asymptotic cones of coarse median spaces. This applies to a large class of finitely generated groups, via their Cayley graphs. We show that such groups satisfy the rapid decay property. We also recover the result of Behrstock, Drutu and Sapir, that the asymptotic cone of the mapping class group embeds in a finite product of trees.
2010 Mathematics Subject Classification : 20F65
Geom. Dedicata 170 (2014) 157--176.