# Bisectors in symmetric spaces

In William Goldman's book Complex Hyperbolic Geometry, bisector hypersurfaces play an important role. Given two points $x,y$, the bisector is the set of points equidistant from $x$ and from $y$. Do they also play an important role in the study of higher rank symmetric spaces, and do they admit concrete descriptions, say in the Siegel upper half-plane? References are especially welcome.

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One of the interest of bisectors is to prove that some non-arithmetic groups of isometries are nonetheless discrete by constructing fundamental domains for their action. Higher rank spaces are much more rigid, and all lattices are arithmetic. This does not mean that bissectors are useless, but their purpose seems less obvious, and it will probably be more difficult to find relevant literature. –  Benoît Kloeckner Nov 18 '12 at 11:33
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. –  user28191 Nov 18 '12 at 19:55
In higher rank the most useful bisectors appear in the case of $SL(n)$ where instead of the Riemannian metric one uses a non -symmetric metric defined by Selberg in his 1960 paper. The key is that Selberg's bisectors are linear, unlike the Riemannian ones. –  Misha Nov 19 '12 at 7:09
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. –  user28191 Nov 19 '12 at 10:31
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. –  user28191 Nov 20 '12 at 14:04