The structure of primitive permutation groups with finite suborbits and t.d.l.c. groups admitting a compact open subgroup that is maximal

Abstract

This paper is about the structure of infinite primitive permutation groups and totally disconnected locally compact groups ("tdlc groups'"). The permutation groups we investigate are subdegree-finite (i.e. all orbits of point stabilisers are finite). Automorphism groups of connected, locally finite graphs are examples of subdegree-finite permutation groups. The tdlc groups we investigate all have a maximal subgroup that is compact and open. Tdlc groups with few open subgroups (recently studied by Pierre-Emmanuel Caprace and Timoth\'ee Marquis) are examples of such groups. We prove a classification result, and use it to show that every closed, subdegree-finite primitive group is a primitive subgroup of a product H Wr F_1 [X] F_2 Wr ... [X] F_{m-1} Wr F_m, where H is a closed, subdegree-finite and primitive group that is either finite or one-ended and almost topologically simple. The groups F_i are transitive and finite and m is finite. The product [X] here denotes the recently discovered box product, and all wreath products here act via their product action. We apply this permutational result to tdlc groups. If G is a tdlc group then it contains a compact open subgroup V. The permutation group induced by the action of G on the coset space G/V is called the Schlichting completion of the pair (G,V), and is denoted G//V. Knowledge of this action of G on G/V underpins many recent influential results in tdlc theory. We show that if V is a maximal subgroup of G, and G is non-compact, then G//V is subject to a topological interpretation of our result for primitive groups. We use this to show that in this case, if G//V is nondiscrete, preserves no nontrivial homogenous cartesian decomposition on G/V, and does not split nontrivially over a compact open subgroup, then the monolith of G//V is a nondiscrete, one-ended, topologically simple, compactly generated tdlc group.