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The second bounded cohomology of hypo-Abelian groups. (English) Zbl 1143.55003
The definition of the bounded cohomology of a group \(G\) is very similar to that of its ordinary cohomology. It is defined as follows. For each positive integer \(n\), let
\[ B^n(G) = \{ f : G^n \longrightarrow \mathbb{R} \; | \; \| f\| < \infty \}, \] where
\[ \| f\| = \text{sup}\{| f(x_1, x_2, \dots , x_n)| , (x_1, x_2, \dots, x_n) \in G^n\}. \] The bounded cohomology \(\widehat H^*(G)\) of \(G\) is the homology of the complex \[ 0 \longrightarrow \mathbb{R} \overset{ 0} \longrightarrow B^1(G) \overset{d_1} \longrightarrow B^2(G) \overset {d_2}\longrightarrow B^3(G) \overset {d_3}\longrightarrow \dots, \] where \[ \begin{split} d_n(f)(g_1, \dots, g_{n+1}) = f(g_2, \dots g_{n+1}) + \sum_{i=1}^n (-1)^i f(g_1, \dots, g_ig_{i+1}, \dots g_{n+1}) \\ + \, (-1)^{n+1} f(g_1, \dots g_n).\end{split} \] Thus, the only difference between bounded and ordinary cohomologies of \(G\) is that in the bounded case we insist that our cocycles and coboundaries be bounded with respect to the sup norm.
It is easy to see that \(\widehat H^1(G)\) consists of all bounded homomorphisms from \(G\) to \(\mathbb{R}\). Since there are no such homomorphisms unless \(G\) is trivial, it follows that \(\widehat H^1(G) = 0\). Therefore, \(\widehat H^2(G)\) has to be investigated. K. Fujiwara [The second bounded cohomology of a group with infinitely many ends, math.GR/9505208] conjectured that if \(\widehat H^2(G)\) is non-zero, then it has to be an infinite-dimensional vector space over \(\mathbb{R}\). However, as pointed out in the paper, there are some linear groups for which \(\widehat H^2(G)\) is non-zero but finite-dimensional. All the known counterexamples to Fujiwara’s conjecture are linear groups, and linear groups contain perfect subgroups. (Recall that a group is perfect if it is equal to the subgroup generated by its commutators.) Motivated by this observation, the author investigates Fujiwara’s conjecture for finite groups which do not contain any non-trivial perfect subgroup. These are called Hypo-Abelian groups. In the main theorem, the author proves Fujiwara’s conjecture for Hypo-Abelian groups using the transfinitely extended derived series for \(G\).
MSC:
55N35 Other homology theories in algebraic topology
20J06 Cohomology of groups
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References:
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