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Thanks. I'll think about this. I have a family of examples where I'm pretty sure the Riemannian bisectors will provide walls for a fundamental domain, but I don't know how to check this because I don't know how to explicitly describe them. Selberg's bisectors might work as well and look more computable. I'll think about it. But they look less likely to behave well with respect to maps between symmetric spaces, no?
OK. I looked at Selberg's paper. I've never seen something like that before. I think that I'm still interested in the Riemannian metric, but this Selberg thing could be useful too. Does it appear commonly? Are there other more extended references? He talks about one case, SL(n), and doesn't say very much.
Thanks. This is Selberg's paper On discontinuous groups...'? I'll see if I can get a hold of that. I'm not very worried if the bisectors are linear or not. I just want to build a fundamental domain for a group generated by some translations in the Siegel upper half-plane, where by translation I mean $Z \mapsto Z + B$`, $B$ a symmetric matrix.
One might still be interested in constructing a nice fundamental domain for a discrete group, even if one already knows a priori whether the group is arithmetic.