Why does not a closed 3-manifold modelled on SL(2,R) admit a metric of nonpositive curvature? I was reading the paper `actions of discrete groups on nonpositively curved spaces' written by Kapovich and Leeb.
In this paper, they proved that generic mapping class groups are not Hadamard groups, i.e. no discrete actions on CAT(0) spaces by semi-simple isometries.
In their proof, they said that since the unit tangent bundle of a hyperbolic surface is modelled on SL(2,R)-geometry, it does not admit a metric of nonpositive curvature.
How can I prove the above statement? Can you suggest me any references?
 A: I am not sure that this is what the OP is looking for, but here is a justification of the fact that the unit tangent bundle of a hyperbolic surface cannot be endowed with a nonpositively curved metric.
Let $\Sigma$ be a closed surface of genus $\geq 2$. The unit tangent bundle $\mathrm{UT}(\Sigma)$ naturally projects onto $\Sigma$, each fiber being a circle $\mathbb{S}^1$:
$$\mathbb{S}^1 \to \mathrm{UT}(\Sigma) \to \Sigma \hspace{1cm} (1)$$
This observation yields a short exact sequence
$$1 \to \mathbb{Z} \to \pi_1(\mathrm{UT}(\Sigma)) \to \pi_1(\Sigma) \to 1.$$
In particular, the fundamental group of the $3$-manifold $\mathrm{UT}(\Sigma)$ contains a normal infinite cyclic subgroup $\langle a \rangle$. Assume for a moment that $\mathrm{UT}(\Sigma)$ can be endowed with a nonpositively curved metric. So $\pi_1(\mathrm{UT}(\Sigma))$ is a CAT(0) group, and it must contain a finite-index subgroup admitting $\langle a \rangle$ as a direct factor, say $\langle a \rangle \times K$. Let $\Xi \to \mathrm{UT}(\Sigma)$ denote the corresponding finite cover. Then, by construction, $\Xi$ is homeomorphic to some product $\mathbb{S}^1 \times M$, whose decomposition is compatible with (1). In other words, the fiber bundle (1) is trivial up to a finite cover, which is clearly false.
The arguments are essentially extracted from the proof of Theorem II.7.27 from Bridson and Haefliger's book Metric spaces of non-positive curvature.
A: If you read our paper a bit further, you will find that on page 348 we mention that this result is due to Eberlein and give a reference to his 1982 paper.
More precisely, he proves a more general theorem that a nonpositively curved compact Riemannian manifold whose fundamental group has nontrivial center has a finite-sheeted covering space which splits smoothly as a product of a torus with another factor. For Seifert manifolds, this is equivalent to the types $E^3$ and $H^2\times R$.
