# zbMATH — the first resource for mathematics

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
Full Text: