Tessellations of solvmanifolds

Witte, Dave

doi:10.1090/S0002-9947-98-01980-1

Tessellations of solvmanifolds
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Trans. Amer. Math. Soc. 350 (1998), 3767-3796 Request permission

Abstract:

Let $A$ be a closed subgroup of a connected, solvable Lie group $G$, such that the homogeneous space $A\backslash G$ is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation $A\backslash G/\Gamma$ of $A\backslash G$ is finitely covered by a compact homogeneous space $G’/\Gamma ’$. We prove that the covering map can be taken to be very well behaved — a “crossed" affine map. This establishes a connection between the geometry of the tessellation and the geometry of the homogeneous space. In particular, we see that every geometrically-defined flow on $A\backslash G/\Gamma$ that has a dense orbit is covered by a natural flow on $G’/\Gamma ’$.

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