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**1 - 3**of**3**### Invariant random subgroups . . .

, 2014

"... An invariant random subgroup (IRS) of a countable discrete group Γ is, by definition, a conjugation invariant probability measure on the compact metric space Sub(Γ) of all subgroups of Γ. We denote by IRS(Γ) the collection of all such invariant measures. Theorem 0.1. Let Γ < GLn(F) be a countab ..."

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An invariant random subgroup (IRS) of a countable discrete group Γ is, by definition, a conjugation invariant probability measure on the compact metric space Sub(Γ) of all subgroups of Γ. We denote by IRS(Γ) the collection of all such invariant measures. Theorem 0.1. Let Γ < GLn(F) be a countable non-amenable linear group with a simple, center free Zariski closure. There exits a non-discrete group topology St on Γ such that for every µ ∈ IRS(Γ), µ-almost every subgroup 〈e 〉 6 = ∆ ∈ Sub(Γ) is open. Moreover there exits a free subgroup F < Γ with the following properties: • F ∩ ∆ is an infinitely generated free group, for every open subgroup

### THE TOPOLOGY OF INVARIANT RANDOM SURFACES

"... Abstract. We study the topological type of a generic surface, with respect to a unimodular measure on the space of pointed hyperbolic surfaces. Uni-modularity is equivalent to saying that the measure comes from an invariant random subgroup of the Lie group of isometries of the hyperbolic plane. 1. ..."

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Abstract. We study the topological type of a generic surface, with respect to a unimodular measure on the space of pointed hyperbolic surfaces. Uni-modularity is equivalent to saying that the measure comes from an invariant random subgroup of the Lie group of isometries of the hyperbolic plane. 1.

### GENERIC IRS IN FREE GROUPS, AFTER BOWEN

"... Abstract. Let E be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space (X,B, µ). Let ([E], du) be the the (Polish) full group endowed with the uniform metric. If Fr = 〈s1,..., sr 〉 is a free group on r-generators and α ∈ Hom(Fr, [E]) t ..."

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Abstract. Let E be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space (X,B, µ). Let ([E], du) be the the (Polish) full group endowed with the uniform metric. If Fr = 〈s1,..., sr 〉 is a free group on r-generators and α ∈ Hom(Fr, [E]) then the stabilizer of a µ-random point α(Fr)x 2 Fr is a random subgroup of Fr whose distribution is conjugation invariant. Such an object is known as an invariant random subgroup or an IRS for short. Bowen’s generic model for IRS in Fr is obtained by taking α to be a Baire generic element in the Polish space Hom(Fr, [E]). The lean aperiodic model is a similar model where one forces α(Fr) to have infinite orbits by imposing that α(s1) be aperiodic. In this setting we show that for r < ∞ the generic IRS α(Fr)x 2 Fr is of finite index a.s. if and only if E = E0 is the hyperfinite equivalence relation. For any ergodic equivalence relation we show that a generic IRS coming from the lean aperiodic model is co-amenable and core free. Finally, we consider the situation where α(Fr) is highly transitive on almost every orbit and in particular the corresponding IRS is supported on maximal subgroups. Using a result of Le-Mâıtre we show that such examples exist for any aperiodic ergodic E of finite cost. For the hyper-finite equivalence relation E0 we show that high transitivity is generic in the lean aperiodic model. 1.