The second bounded cohomology of hypo-Abelian groups.

*(English)*Zbl 1143.55003The 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 \(G\).

\[ B^n(G) = \{ f : G^n \longrightarrow \mathbb{R} \; | \; \| f\| < \infty \}, \] 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 \(G\).

Reviewer: Sunil Chebolu (Illinois)

Full Text:
DOI

##### References:

[1] | Fujiwara, K., The second bounded cohomology of a group with infinitely many ends |

[2] | Grigorchuk, R., Some results on bounded cohomology, London math. soc. lecture note ser., 202, 111-163, (1993) · Zbl 0853.20034 |

[3] | Gromov, M., Volume and bounded cohomology, Publ. math. inst. hautes etudes sci., 56, 5-100, (1982) · Zbl 0516.53046 |

[4] | Ivanov, N., Foundation of theory of bounded cohomology, J. soviet math., 37, 1090-1114, (1987) · Zbl 0612.55006 |

[5] | Monod, N., Continuous bounded cohomology of locally compact groups, Lecture notes in math., vol. 1758, (2001), Springer · Zbl 0967.22006 |

[6] | Matsumoto, S.; Morita, S., Bounded cohomology of certain groups of homomorphisms, Proc. amer. math. soc., 94, 539-544, (1985) · Zbl 0536.57023 |

[7] | Robinson, D., A course in the theory of groups, (1995), Springer New York |

[8] | Rudin, W., Functional analysis, (1973), McGraw-Hill New York · Zbl 0253.46001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.