# 3-manifold Groups and Nonpositive Curvature by Kapovich M. By Kapovich M.

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Example text

32 (and keeping the same notation), U ◦ j = Id. 61). 19) for a Dirac realization, it follows that i is injective. By a dimension argument, it follows that the sequence is exact. We now concentrate on constructing the quasi-Poisson bivector ﬁeld π . 20, we have the following. 35. π is well deﬁned. Proof. 29). Hence it must be in the image of i. More explicitly, we ﬁnd that there exists an X such that ∗ (X, C ∗ (α)) ∈ L, ρ ∨ (dJ (X) + (σ ∨ )∗ ρM (α)) = 0. 31) to replace CρM , we ﬁnd that ∗ σ ∗ (dJ (X) + (σ ∨ )∗ ρM (α)) = 0.

Since (iii) and the Leibniz identity for [·, ·] are together equivalent to (iv), it follows that (i)–(iv) are equivalent to each other. 7. If (M, π ) is a quasi-Poisson g-manifold, then the generalized distribution π (α) + ρM (v) ⊆ T M, for α ∈ T ∗ M, v ∈ g, is integrable. This result shows that the singular distribution discussed in [2, Thm. 2] in the context of Hamiltonian quasi-Poisson manifolds is integrable even without the presence of a moment map (and without the positivity of (·, ·)g ).

There is a one-to-one correspondence between presymplectic realizations of G endowed with the Cartan–Dirac structure, and quasi-Hamiltonian gmanifolds. 15, we conclude that general Dirac realizations of Cartan–Dirac structures must be “foliated’’ by quasi-Hamiltonian gmanifolds. Since Hamiltonian quasi-Poisson manifolds, in the sense of , also have this property [2, Sec. 10], we are led to investigate the relationship between these objects. 1 The equivalence theorem For a quasi-Poisson g-manifold (M, π), a momentum map is a g-equivariant map J : M −→ G (with respect to the inﬁnitesimal action by conjugation on G) satisfying the condition [2, Lem.