There are various definitions of boundary of a hyperbolic group. Which of those generalize to semi-hyperbolic groups (in the sense of Alonso and Bridson)? 


The example I have in mind is a semisimple group over a nonArchemedian local field.The answer I'd like to get is something like a nonHausdorff space which is a union of partial flag manifolds $G/P$ (which can be described as the quotient of the complement to $G$ in the De Concini-Procesi compactification by the right action of $G$).



boundary of semihyperbolic groups
I don't think any notion of boundary has been developed specifically for semi-hyperbolic groups, but there have been some recent generalizations of boundaries of hyperbolic groups that may be of interest to you.


1. Charney and Sultan defined the *contracting boundary* of a CAT(0) group ([arXiv:1308.6615](http://arxiv.org/abs/1308.6615)). (I don't know enough about your setting to know whether your groups act nicely on some sort of CAT(0) building, which might help.)
2. Charney's student Cordes generalized the contracting boundary to define the *Morse boundary* of *any* finitely generated group ([arXiv:1502.04376](http://arxiv.org/abs/1502.04376)).


Note that both of these boundaries can be, and frequently are, empty. I think they're also both always Hausdorff, so may not give exactly the answer you want. But hopefully it's somewhere to start.


