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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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  • $\begingroup$ 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. $\endgroup$
    – Benoît Kloeckner
    Nov 18, 2012 at 11:33
  • $\begingroup$ 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. $\endgroup$
    – user28191
    Nov 18, 2012 at 19:55
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    $\begingroup$ 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. $\endgroup$
    – Misha
    Nov 19, 2012 at 7:09
  • $\begingroup$ 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. $\endgroup$
    – user28191
    Nov 19, 2012 at 10:31
  • $\begingroup$ 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. $\endgroup$
    – user28191
    Nov 20, 2012 at 14:04

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Although I cannot address your specific question, of whether bisectors "play an important role in the study of higher rank symmetric spaces," I can say that bisectors are the essence of Voronoi diagrams. So the 2009 paper by Frank Nielsen and Richard Nock, entitled "Hyperbolic Voronoi diagrams made easy" (arXiv link), could well be relevant:

We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. [...]


       Hyperbolic VorDiag

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    $\begingroup$ That's a nice reference, but hyperbolic space is Rank 1, which means, in particular, that there are totally geodesic hyperplanes, which are the bisectors. In higher rank, there are no totally geodesic hyperplanes, so the whole theory (to the extent it exists at all) is \emph{completely} different. $\endgroup$
    – Igor Rivin
    Nov 18, 2012 at 3:05
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    $\begingroup$ Also cool pictures. But yes, I'm particularly interested in higher rank, especially the Siegel upper half-plane. $\endgroup$
    – user28191
    Nov 18, 2012 at 9:29

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