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Existence of non-Banach bounded cohomology. (English) Zbl 0893.55004
For any discrete group \(G\) and \(n\in\mathbb{Z}\) the bounded cohomology group \(H_b^n(G;\mathbb{R})\) admits a canonically defined pseudonorm \(\|\cdot\|\). The paper gives a negative answer to the question whether \(\|\cdot\|\) is a norm, i.e. \((H_b^n (G;\mathbb{R}),\|\cdot\|)\) is a Banach space. The author shows that \(H_b^3(\mathbb{Z}* \mathbb{Z},\mathbb{R},\|\cdot\|)\) is not a Banach space (Theorem 1). Moreover, for any discrete group \(G\) admitting a surjective homomorphism \(f:G\to \mathbb{Z} * \mathbb{Z}\), \((H_b^3(G, \mathbb{R}),\|\cdot \|)\) is not a Banach space (Corollary). The author establishes that for \(n\geq 5\) there exists a finitely generated discrete group \(G\) such that \((H_b^n (G;\mathbb{R}),\|\cdot\|)\) is not a Banach space (Theorem 2). The proofs are very dense and use some interesting arguments of hyperbolic geometry as well as a result of S. Matsumoto and S. Morita.

MSC:
55N35 Other homology theories in algebraic topology
57M07 Topological methods in group theory
20J05 Homological methods in group theory
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