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# Lower bounds for covolumes of arithmetic lattices in $$PSL_2(\mathbb R)^n$$

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### Abstract

We study the covolumes of arithmetic lattices in $$PSL_2(\mathbb R)^n$$ for $$n\geq 2$$ and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let $$\mu$$ be the Euler-Poincar\'e measure on $$PSL_2(\mathbb R)^n$$ and $$\chi=\mu/2^n$$. We show that the Hilbert modular group $$PSL_2(\mathfrak o_{k_{49}})\subset PSL_2(\mathbb R)^3$$, with $$k_{49}$$ the totally real cubic field of discriminant $$49$$ has the minimal covolume with respect to $$\chi$$ among all irreducible lattices in $$PSL_2(\mathbb R)^n$$ for $$n\geq 2$$ and is unique such lattice up to conjugation. The uniform lattice of minimal covolume with respect to $$\chi$$ is the normalizer $$\Delta_{k_{725}}^u$$ of the norm-1 group of a maximal order in the quaternion algebra over the unique totally real quartic field with discriminant $$725$$ ramified exactly at two infinite places, which is a lattice in $$PSL_2(\mathbb R)^2$$. There is exactly one more lattice in $$PSL_2(\mathbb R)^2$$ and exactly one in $$PSL_2(\mathbb R)^4$$ with the same covolume as $$\Delta_{k_{725}}^u$$, which are the Hilbert modular groups corresponding to $$\mathbb Q(\sqrt{5})$$ and $$k_{725}$$. The two lattices $$\Delta_{k_{725}}^u$$ and $$PSL_2(\mathfrak o_{\mathbb Q(\sqrt{5})})$$ have the smallest covolume with respect to the Euler-Poincar\'e measure among all arithmetic lattices in $$G_n$$ for all $$n\geq 2$$. These results are in analogy with Siegel's theorem on the unique minimal covolume (uniform and non-uniform) Fuchsian groups and its generalizations to various higher dimensional hyperbolic spaces due to Belolipetsky, Belolipetsky-Emery, Stover and Emery-Stover.

### Author and article information

###### Journal
26 January 2015
###### Article
1501.06443
ebb6b058-be40-469b-bf53-f9ca625660c0