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**Some new examples of simple \(p\)-local compact groups.**
*(English)*
Zbl 1437.55021

The theory of \(p\)-local compact groups was introduced by Broto, Levi and Oliver [C. Broto et al., Geom. Topol. 11, 315–427 (2007; Zbl 1135.55008)] to model combinatorially the classifying spaces of \(p\)-compact groups and \(p\)-completed classifying spaces of compact Lie groups. This paper contains the construction of two families of \(p\)-local compact groups which are not of this form.

The construction in the case \(p=3\) uses exotic \(3\)-local finite groups from [A. Díaz et al., Trans. Am. Math. Soc. 359, No. 4, 1725–1764 (2007; Zbl 1113.55010)] which are defined over certain semidirect products \( S_k = (\mathbb{Z}/3^k)^2 \rtimes \mathbb{Z}/3\) for \(k \geq 2\). The fusion systems of these \(3\)-local finite groups are generated by the automorphisms of three subgroups, and these automorphisms can be extended to three corresponding subgroups of the union \(S\) of the groups \(S_k\), which is a discrete \(3\)-toral group of the form \((\mathbb{Z}/3^{\infty})^2 \rtimes \mathbb{Z}/3\). The fusion system over \(S\) generated by these three automorphism groups defines a \(3\)-local compact group which is simple, in the sense that its only normal fusion subsystems are finite.

For each prime \(p \geq 5\), the authors first consider a family of exotic \(p\)-local finite groups constructed in [C. Broto et al., J. Am. Math. Soc. 16, No. 4, 779–856 (2003; Zbl 1033.55010)] over certain semidirect products \((\mathbb{Z}/p^k)^{p-1} \rtimes \mathbb{Z}/p\) which are maximal nilpotency class \(p\)-groups, like the groups \(S_k\) mentioned above. The fusion system of each of these \(p\)-local finite groups contains a saturated fusion system of index two, which determines another exotic \(p\)-local finite group. All these \(p\)-local finite groups are described in detail, including a description of the automorphism groups of three subgroups which generate the fusion systems. Then two \(p\)-local compact groups are constructed for each prime \(p \geq 5\) using these families of \(p\)-local finite groups in the same manner as in the case \(p=3\). As expected from the construction, the fusion system of one of these two \(p\)-local compact groups is a subfusion system of index two of the other one, and it is simple.

It is shown that none of these \(p\)-local compact groups can be realized by a \(p\)-compact group or by a compact Lie group. The paper also contains some auxiliary results which are interesting and useful on their own. For instance, a saturation criterion for fusion systems over discrete \(p\)-toral groups, a classification of saturated fusion subsystems of index prime to \(p\) in this context, and for \(p\)-local compact groups which are realized by a \(p\)-compact group, an identification of the classifying spaces of centralizer \(p\)-local compact groups with mapping spaces.

The construction in the case \(p=3\) uses exotic \(3\)-local finite groups from [A. Díaz et al., Trans. Am. Math. Soc. 359, No. 4, 1725–1764 (2007; Zbl 1113.55010)] which are defined over certain semidirect products \( S_k = (\mathbb{Z}/3^k)^2 \rtimes \mathbb{Z}/3\) for \(k \geq 2\). The fusion systems of these \(3\)-local finite groups are generated by the automorphisms of three subgroups, and these automorphisms can be extended to three corresponding subgroups of the union \(S\) of the groups \(S_k\), which is a discrete \(3\)-toral group of the form \((\mathbb{Z}/3^{\infty})^2 \rtimes \mathbb{Z}/3\). The fusion system over \(S\) generated by these three automorphism groups defines a \(3\)-local compact group which is simple, in the sense that its only normal fusion subsystems are finite.

For each prime \(p \geq 5\), the authors first consider a family of exotic \(p\)-local finite groups constructed in [C. Broto et al., J. Am. Math. Soc. 16, No. 4, 779–856 (2003; Zbl 1033.55010)] over certain semidirect products \((\mathbb{Z}/p^k)^{p-1} \rtimes \mathbb{Z}/p\) which are maximal nilpotency class \(p\)-groups, like the groups \(S_k\) mentioned above. The fusion system of each of these \(p\)-local finite groups contains a saturated fusion system of index two, which determines another exotic \(p\)-local finite group. All these \(p\)-local finite groups are described in detail, including a description of the automorphism groups of three subgroups which generate the fusion systems. Then two \(p\)-local compact groups are constructed for each prime \(p \geq 5\) using these families of \(p\)-local finite groups in the same manner as in the case \(p=3\). As expected from the construction, the fusion system of one of these two \(p\)-local compact groups is a subfusion system of index two of the other one, and it is simple.

It is shown that none of these \(p\)-local compact groups can be realized by a \(p\)-compact group or by a compact Lie group. The paper also contains some auxiliary results which are interesting and useful on their own. For instance, a saturation criterion for fusion systems over discrete \(p\)-toral groups, a classification of saturated fusion subsystems of index prime to \(p\) in this context, and for \(p\)-local compact groups which are realized by a \(p\)-compact group, an identification of the classifying spaces of centralizer \(p\)-local compact groups with mapping spaces.

Reviewer: José Maria Cantarero Lopez (Mérida)