Groups generated by involutions, numberings of posets, and central measures

Abstract

1. Definitions. An infinite countable ordered set {P,≻, ∅} with minimal element ∅ and no maximal elements is called a locally finite poset if all its principal ideals are finite. A monotone numbering of P (or a part of P ) is an injective map φ : N → P from the set of positive integers to P satisfying the following conditions: if φ(n) ≻ φ(m), then n > m, with φ(0) = ∅. The distributive lattice ΓP of all finite ideals of a locally finite poset {P,≻} forms an N-graded graph (the Hasse diagram of the lattice). A monotone numbering of P can be identified in a natural way with a maximal path in the lattice ΓP . The set TP of all monotone numberings of P , that is, the space of infinite paths in the graph ΓP can be endowed with the natural structure of a Borel and topological space. In the terminology related to the Young graph, the poset P is the set of Z+-finite ideals, that is, Young diagrams, and monotone numberings are Young tableaux. Let P be a finite (|P | < n ∈ N) or locally finite (n = ∞) poset. For each i < n, we define an involution σi acting correctly on the space of numberings TP = {φ} of P :