Can you please help me with the following:
Let $M$ be a compact, connected Riemannian manifold with convex boundary of nonpositive curvature and dimension $n$. Let $\tilde{M}$ be its universal cover and $\Gamma$ be the group of deck translations. Let $A \subset \Gamma$ be a free abelian subgroup of rank $n-2$. Denote by $H_A$ the closed convex hull of the union of all $A$-invariant $\Gamma$-flats (By a $\Gamma$-flat $X$ I mean a closed flat hypersurface of codimension $1$ such that the subgroup of $\Gamma$ that fixes $X$ acts cocompactly on it). Can you tell me why the following is true:
1. $H_A$ is a subset of the set $MIN(A)$ of minimal displacements, where a minimal displacement is a point such that $d(x, \phi(x))$ is minimal among all $x \in \tilde{M}$ for every $\phi \in A$.
2. $H_A$ has the form $Z \times \mathbb{R}$ for some closed manifold $Z$.
3. The quotient of $H_A$ by the normalizer of $A$ in $\Gamma$ is a Seifert manifold.
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$\begingroup$ A little context would help. Why do you think these things are true? $\endgroup$– HJRWJun 4, 2012 at 2:26
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$\begingroup$ Actually, 1 is false, the inclusion is the other way. You also have to exclude flat manifolds. $\endgroup$– MishaJun 4, 2012 at 2:31
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$\begingroup$ @philipp: You should read few relevant books/papers on manifolds/spaces of nonpositive curvature: 1. P.Eberlein "Geometry of Nonpositively Curved Manifolds", 2. M.Bridson, A.Haefliger, "Metric Spaces of Non-Positive Curvature", 3. W.Ballmann "Lectures on spaces of nonpositive curvature", 4. V.Schroeder, "Codimension one tori in manifolds of nonpositive curvature," 5. M. Kapovich, B. Kleiner, "Weak hyperbolization conjecture for 3-dimensional CAT(0) groups." Once you have sufficient background in NPC/CAT(0) geometry, you can answer your questions yourself. $\endgroup$– MishaJun 4, 2012 at 4:01
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