On Axiom H (Q1590943)

\textit{M. Bestvina} [in Mich. Math. J. 43, No. 1, 123-139 (1996; Zbl 0872.57005)], considered an object he called a \(Z\)-structure associated to a group \(G\), which plays the role of a boundary of \(G\). He introduced a notion called Axiom H, which can be satisfied by a \(Z\)-structure. Bestvina stated (and proved) the fact that if \(G\) is a hyperbolic group and \(Z\) its Gromov boundary, then Axiom H holds for \(Z\). This is stated as Proposition 1.18 in Bestvina's paper. In the paper under review, the author shows that this proposition of Bestvina is false. He proves that Axiom H is not satisfied when \(Z\) is the Gromov boundary of the free group of rank 2. He proves a weaker fact than Bestvina's Proposition 1.18, which says that almost all points of the Gromov boundary satisfy Axiom H. Bestvina used Proposition 1.18 to obtain some results relating the global and local Steenrod homology of Gromov boundaries of hyperbolic groups. In the paper under review, the author shows that slightly weaker results than those stated by Bestvina hold. In fact, he proves the following Theorem: Let \(Z\) be the Gromov boundary of a hyperbolic group. Then, for all \(q\geq 0\), one of the following holds for Steenrod homology with coefficients in a countable field. (1) For all \(z\in Z\), the natural map \(H_q(Z)\to H_q(Z,Z-\{z\})\) is an isomorphism and the two vector spaces are finite-dimensional. (2) \(H_q(Z,Z-\{z\})\) is uncountable for some \(z\in Z\). Bestvina had a statement of the same kind (Proposition 1.17), stating that with the same hypotheses, then \textit{for all} \(z\in Z\), one of the two properties is satified : (1) The natural map \(H_q(Z)\to H_q(Z,Z-\{z\})\) is an isomorphism and the two vector spaces are finite-dimensional. (2) \(H_q(Z,Z-\{z\})\) is uncountable. In this paper, this is asked now as an open question.