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There is existing literature on fundamental domains for Hilbert-Blumenthal surfaces, perhaps most noticeably what Siegel did. I am interested in whether such fundamental domains have been approached using Dirichlet domains. More specifically...

I want to know what a Dirichlet domain for a manifold covering a Hilbert modular variety looks like. What kind of symmetry does it have? Is it geometrically finite? What topologically invariant properties does it have (independent of center)? Is there a useful analogy or method to visualize it? If you already know the answer to this or know of a good reference for it, you probably don't need to read the rest of my question -- unless the question doesn't make sense in which case you can tell me where my construction falls apart.

Let $K$ be a totally real number field, let $\mathbb{Z}_K$ be its ring of integers, let $r=[K:\mathbb{Q}]>1$. Let $\Gamma=\mathrm{PSL}_2(\mathbb{Z}_K)$ and let $\mathcal{H}^2$ be the upper half-space model for the hyperbolic plane. Then there is a discrete action by isometries $$\Gamma\times(\mathcal{H}^2\times\overset{r}{\cdots}\times\mathcal{H}^2)\rightarrow(\mathcal{H}^2\times\overset{r}{\cdots}\times\mathcal{H}^2)\\ \big(\gamma,(p_1,\dots,p_r)\big)\mapsto \big(\sigma_1(\gamma)(p_1),\dots,\sigma_r(\gamma)(p_r)\big),$$ where the $\sigma_\ell$ are the $r$ Galois automorphisms applied to the entries of $\gamma$, and $\sigma(\gamma_\ell)(p_\ell)$ is the action by Möbius transformation. Then $(\mathcal{H}^2\times\overset{r}{\cdots}\times\mathcal{H}^2)/\Gamma$ is a Hilbert modular variety. Also, from Selberg, we know that $\Gamma$ contains a maximal torsion-free subgroup $\Delta$. Let $M=(\mathcal{H}^2\times\overset{r}{\cdots}\times\mathcal{H}^2)/\Delta$, then $M$ is a $2n$-dimensional manifold.

Now let $p\in\mathcal{H}^2\times\overset{r}{\cdots}\times\mathcal{H}^2$. We find a Dirichlet domain for $M$ by looking at the set of geodesics that connect $p$ to each image in the orbit $\Delta(p)$, then taking a perpendicular bisector of each of these geodesics. We know that $V$ has finite volume and finitely many cusps, and $M$ is a finite cover of $V$ so the same is true for $M$. Also, since $\Delta$ is torsion-free, its only fixed points are on the boundary. Thus there will be a region containing $p$ enclosed by the set of perpendicular bisectors, except for some finite number of points on the boundary where they meet. This region is the Dirichlet domain for $M$ centered at $p$, which I'll denote by $D(p)$.

To make the construction more canonical, let's just take $p=(i,\dots,i)$ (which in each coordinate, in the upper half-space model, is one unit up the half-axis, above $0$ which lies in $\partial\mathcal{H}^2$). What can we say about $D(p)$? What kind of symmetry does it have? How many faces does it have?

We know that the projection of the group action onto any one factor is dense, yet the action on the product is discrete because any time an orbit accumulates in one factor, it will diverge to $\infty$ in another. So it does not seem very helpful to try to conceive of a Dirichlet domain by thinking of it as a product of $2$-dimensional ones. What is a good way to conceive of $D(p)$? Perhaps take $r=2$ and just look at Hilbert-Blumenthal surfaces.


Edit - March 14, 2017

This has been up for about a week with no answers. So, I was wondering then whether the answer to this question is "known," i.e. does there exist a publication where this question is answered?

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    $\begingroup$ I think it's extremely hard to compute Dirichlet domains in general, especially in cases where bisectors are not totally geodesic (so all cases of nonconstant curvature). The only work I'm aware of on this topic outside of real hyperbolic spaces is on complex hyperbolic surfaces, see eg. ams.org/mathscinet-getitem?mr=2888244 for something analogue to your question. $\endgroup$ – Jean Raimbault Mar 23 '17 at 8:18
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    $\begingroup$ To complement Jean's remark about complex hyperbolic space: you can also look at the work of Martin Deraux on this case. Also, you may know already that there are constructions of Voronoi that provide nice fundamental domains for $\mathrm{PGL}_2(\mathbb{Z}_K)$ acting on $\mathcal{H}^r\times \mathbb{R}^{r-1}$; you may want to look at papers of Paul Gunnells on this, but those are not Dirichlet domains! $\endgroup$ – Aurel Mar 23 '17 at 11:22
  • $\begingroup$ @JeanRaimbault That article looks very useful, thanks. I'm going to take a more careful look. $\endgroup$ – j0equ1nn Mar 23 '17 at 23:34
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    $\begingroup$ Presumably you're using the product Riemannian metric, but it might also be worth considering a Finsler metric which is an l^1 sum of the metrics on each factor (or maybe take the l^∞ metric). $\endgroup$ – Ian Agol Mar 29 '17 at 2:31
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    $\begingroup$ @j0equ1nn en.wikipedia.org/wiki/Finsler_manifold?wprov=sfsi1 A Finsler manifold has a path metric induced from a norm on the tangent space. So an l^1 sum of Riemannian metrics gives a Finsler metric. $\endgroup$ – Ian Agol Mar 31 '17 at 3:13

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