Invariant random subgroup

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Definition

For a locally compact group, compatible with the Chabauty topology

Suppose G is a locally compact group. Denote by \mathcal{C}(G) the collection of closed subgroups of G, equipped with the Chabauty topology.

An invariant random subgroup (IRS) on G is a probability measure on the set \mathcal{C}(G) satisfying the following two conditions:

  1. It is a Borel measure with respect to the Chabauty topology, i.e., all Borel subsets under the Chabauty topology are measurable. This makes it a random subgroup.
  2. It is invariant under the action of G by conjugation.

Note on topological constraints

Note that although one may be tempted to define the concept of invariant random subgroup much more generally as simply any probability measure on any collection of subgroups satisfying (2), the particular theory developed for IRSs in the literature heavily relies on the constraints created by the Chabauty topology.

Examples

  • Any single closed normal subgroup of a locally compact group can be used to define an invariant random subgroup with all the measure concentrated on that subgroup.
  • Any closed subgroup that is also an almost normal subgroup (i.e., it has only finitely many conjugates) in a locally compact group can be used to define an invariant random subgroup with the measure uniformly distributed across all the conjugate subgroups.
  • More generally, given any finite collection of conjugacy classes of closed almost normal subgroups, we can assign probabilities to each conjugacy classes to add up to 1, and then divide the probability associated to each conjugacy class equally within the members of that conjugacy class.

References