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Nilpotent \(n\)-tuples in \(\mathrm{SU}(2)\). (English) Zbl 1468.22022

When \(G\) is a topological group and \(\Gamma\) is a finitely generated group, the space of group homomorphisms \(\operatorname{Hom}(\Gamma,G)\) can be regarded as a topological subspace of \(G^\Gamma\). This paper considers \(\operatorname{Hom}(\Gamma,G)\) when \(\Gamma\) is a finitely generated nilpotent group and \(G=\mathrm{SU}(2)\). The case when \(\Gamma=\mathbb{Z}^n\) has been studied by multiple authors, for instance in [A. Adem and F. R. Cohen, Math. Ann. 338, No. 3, 587–626 (2007; Zbl 1131.57003); T. Baird et al., Ill. J. Math. 55, No. 3, 805–813 (2011; Zbl 1278.55027); D. Kishimoto and M. Takeda, Adv. Math. 386, Article ID 107809, 43 p. (2021; Zbl 1478.57040)]. The authors identify the connected components of \(\operatorname{Hom}(\Gamma,\mathrm{SU}(2))\) and their number by noticing that all non-abelian nilpotent subgroups of \(\mathrm{SU}(2)\) are conjugate to a group of generalized quaternions. For instance, if \(F_n\) is the free group on \(n\) symbols and \(\Gamma^{q}_n\) stands for the \(q\)-{th} stage of the central series of \(F_n\), then \(\operatorname{Hom}(F_n/\Gamma^{q}_n,\mathrm{SU}(2))\) has one connected component homeomorphic to \(\operatorname{Hom}(\mathbb{Z}^n,\mathrm{SU}(2))\) and all the remaining homeomorphic to \(\mathrm{PU}(2)\) (provided \(q\geq 3\)).
The spaces \(\{\operatorname{Hom}(F_n/\Gamma^{q}_n,G) \}_{n\geq 0}\) form a simplicial space with realization denoted by \(B(q,G)\), or also \(B_{com}G\) when \(q=2\). The space \(B(q,G)\) is the classifying space of principal \(G\)-bundles of transitional nilpotency class less than \(q\) (see [A. Adem et al., Math. Proc. Camb. Philos. Soc. 152, No. 1, 91–114 (2012; Zbl 1250.57003); Algebr. Geom. Topol. 17, No. 2, 869–893 (2017; Zbl 1360.55003)]). As a product of their work, the authors are able to obtain the mod-2 cohomology ring of \(B_{com} (Q_{2^r})\), where \(Q_{2^r}\) is a quaternion group of order \(2^r\). They also show that the inclusions \(B_{com}\mathrm{SU}(2) \subset B(3, \mathrm{SU}(2)) \subset \cdots \subset B(q, \mathrm{SU}(2)) \subset \cdots\) are homology isomorphisms with coefficients over a ring where 2 is invertible.

MSC:

22E99 Lie groups
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology

Software:

SINGULAR
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References:

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