Citation Context

...invariant under conjugation is called an invariant random subgroup or IRS. One of the goals of this research is to classify the distributions of IRS’s for interesting groups. For example Stuck-Zimmer =-=[SZ94]-=- proved that every ergodic IRS of a higher rank simple Lie group is induced from a lattice subgroup. A complete classification of IRS’s of the infinite symmetric group has been obtained by A. Vershik ...

Citation Context

....3. If Γ does not have property (T) then IRSΓ(N) is a Poulsen simplex. If Γ has property (T) then IRSΓ(N) is a Bauer simplex. The proof of Theorem 1.3 is based on the proof of B. Weiss and E. Glasner =-=[GW97]-=- that the simplex of invariant probability measures for the shift-action of G on {0, 1}G is a Poulsen simplex if G does not have Kazhdan’s property (T) and is a Bauer simplex if G has property (T). Ob...

Citation Context

...which is ergodic but not strongly ergodic. This implies the existence of a sequence {Ai} ∞ i=1 of Borel subsets of X such that • limi→∞ λ(Ai △ gAi) = 0 for every g ∈ Γ; • λ(Ai) = 1/2 for every i, see =-=[JS87]-=-, lemma 2.3. Let Ψn : Sub(N)× Sub(N)×X → Sub(N) be the map Ψn(H1, H2, x) = (H1 ∩ ⊕Jn(x)F )× (H2 ∩ ⊕Kn(x)F ) where Jn(x) = {g ∈ Γ : x ∈ gAn}, Kn(x) = {g ∈ Γ : x ∈ X \ gAn}. Note that Ψn is Γ-equivarian...

Citation Context

... proved that every ergodic IRS of a higher rank simple Lie group is induced from a lattice subgroup. A complete classification of IRS’s of the infinite symmetric group has been obtained by A. Vershik =-=[Ve11]-=-. Many interesting results and applications of IRS’s to semisimple Lie groups are announced in [AB+11] and obtained in [AB+12]. An important subspace of the space of Schreier graphs of a group of inte...

Citation Context

...enough to show that (1/2)(µ1 + µ2) is a limit of ergodic measures. For the definition of weak mixing and strong mixing we recommend [Sch84]. A result mentioned there in combination with a result from =-=[BV93]-=- shows, as is indicated in [GW97], Theorem 2, that there exists a weakly mixing measure-preserving action Γy(X, λ) on standard probability space (X, λ) which is ergodic but not strongly ergodic. This ...

Citation Context

...is convex. So let µ1, µ2 ∈ IRSΓ(N) be ergodic measures. Hence it is enough to show that (1/2)(µ1 + µ2) is a limit of ergodic measures. For the definition of weak mixing and strong mixing we recommend =-=[Sch84]-=-. A result mentioned there in combination with a result from [BV93] shows, as is indicated in [GW97], Theorem 2, that there exists a weakly mixing measure-preserving action Γy(X, λ) on standard probab...

Citation Context

...re two distinguished classes of simplices: a Poulsen simplex is any simplex whose extreme points form a dense subset while a Bauer simplex is any simplex whose extreme points form a closed subset. By =-=[LOS78]-=-, there is a unique Poulsen simplex up to affine isomorphism. Moreover, its set of extreme points is homeomorphic to the Hilbert space ℓ2. For example, the space of all shift-invariant probability mea...

Citation Context

...SO(n, 1)). It follows from A. Vershik’s work [Ve11] that if G is the infinite symmetric group then IRS∗(G) is a Bauer simplex (moreover, its set of extreme points is explicitly parametrized). Also in =-=[Bo12]-=- it is shown that if G is a nonabelian free group then IRS∗(G) is Poulsen (because in this case K(G) is the subspace of all infinite-index subgroups). The idea of the proof of Theorem 1.2 is quite sim...

Citation Context

... of IRS’s to semisimple Lie groups are announced in [AB+11] and obtained in [AB+12]. An important subspace of the space of Schreier graphs of a group of intermediate growth is described by Y.Vorobets =-=[Vo12]-=-. The study of IRS’s is closely related with the study of central characters on groups. Interesting results in this direction were obtained recently by Dudko and Medynets [DM11, DM12] who showed in pa...

Citation Context

... many subgroups. By contrast, the lamplighter groups Ln,p = (Z/pZ) n ≀ Z = ⊕Z(Z/pZ) n ⋊ Z (were ≀ denotes the wreath product) have uncountably many subgroups. Therefore by the result of P. Kropholler =-=[Kro85]-=- any solvable group G of infinite rank also has a nonempty perfect kernel K(G), as it necessarily has a section isomorphic to one of the groups L1,p (p prime) and hence has uncountably many subgroups....

Citation Context

....1. The perfect kernel of Sub(Ln,p) is Sub(An,p) where An,p denotes the subgroup ⊕Z(Z/pZ) n < Ln,p. Moreover, the Cantor-Bendixson rank of Sub(Ln,p) is the first infinite ordinal. The proof relies on =-=[GK12]-=- which gives a comprehensive study of the lattice of subgroups of Ln,p. To explain the next result, recall there are two distinguished classes of simplices: a Poulsen simplex is any simplex whose extr...

Citation Context

...teger d ≥ 1, the simplex of invariant measures for the action of Zd on AZ d by shift transformations is a Poulsen simplex. Observe that the proof of this fact presented in the paper of G. H. Olsen in =-=[Ol79]-=- (which is a good source of information about the Poulsen simplex) has two mistakes, as firstly ergodicity is confused with the mixing property (bottom of page 45) and secondly the statement about the...