×

A product for permutation groups and topological groups. (English) Zbl 1380.20029

Let \(M\leq \mathrm{Sym}(X)\) and \(N\leq \mathrm{Sym}(Y)\) be nontrivial (possibly infinite) permutation groups. The author defines a new product \(\boxtimes\) (the box product) such that \(M\boxtimes N\) is an infinite permutation group and this paper explores some of the product’s interesting properties. Given a tree \(T\) it is possible to partition the vertices into two classes whose vertices are labelled \(M\) and \(N\), respectively, such that no edge joins two vertices with the same label. A subgroup \(G\leq \mathrm{Aut}(T)\) is called locally-\((M,N)\) if each vertex \(v\) labelled \(M\) (respectively, \(N\)) has degree \(\left| Y\right| \) (respectively, \(\left| X\right| \)) and the stabilizer of \(G_{v}\) induces an action which is permutation isomorphic to \(N\) (respectively, \(M\)) on the set of neighbouring vertices. When \(M\) and \(N\) are transitive there exists an essentially unique tree \(T\) and a group \(\mathcal{U}(M,N)\leq \mathrm{Aut}(T)\) which is locally-\((M,N)\) and contains a permutationally isomorphic copy of every locally-\((M,N)\) group. Then, \(M\boxtimes N\;\)is defined to be the permutation group induced by \(\mathcal{U}(M,N)\) on one of the parts of the partition of the vertices of \(T\). The groups \(M\boxtimes N\) and \(\mathcal{U}(M,N)\) are also isomorphic as topological groups under the permutation group topology. This construction generalizes that of the universal group with prescribed local action due to M. Burger and S. Mozes [Publ. Math., Inst. Hautes Étud. Sci. 92, 113–150 (2000; Zbl 1007.22012)] and indeed the group \(U(M)\) defined by Burger and Mozes [loc. cit.] is topologically isomorphic to \(\mathcal{U}(M, \mathrm{Sym}(2))\).
Suppose \(M\) and \(N\) are transitive and \(\mathcal{U}(M,N)\leq \mathrm{Aut}(T)\). Amongst the properties proved are the following: (1) if \(M\) and \(N\) are generated by their point stabilizers, then \(\mathcal{U}(M,N)\) is simple; (2) each locally-\((M,N)\) subgroup \(H\) of \(\mathrm{Aut}(T)\) is conjugate to a subgroup of \(\mathcal{U}(M,N)\); (3) \(M\boxtimes N\) is primitive if and only if \(M\) is primitive and not regular. The author also uses the box product to construct \(2^{\aleph_{0}}\) pairwise nonisomorphic topologically simple groups which are nondiscrete, totally disconnected, locally compact, and compactly generated; this answers a question in [P.-E. Caprace and T. De Medts, Transform. Groups 16, No. 2, 375–411 (2011; Zbl 1235.20026)].

MSC:

20E08 Groups acting on trees
20B07 General theory for infinite permutation groups
20E25 Local properties of groups
22D05 General properties and structure of locally compact groups
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid Link