A 2-generated just-infinite profinite group which is not by Lucchhini A.

By Lucchhini A.

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32 CHAPTER 3 2. , [M, Cor. 7]). As a result, it follows that G + does not admit any nontrivial finite-dimensional unitary representations. Put otherwise, an irreducible unitary representation of G is finite-dimensional if and only if it admits a G + -invariant unit vector. In particular, every ergodic action of G + is weak-mixing, and if it is irreducible, each component is weak-mixing. 3. We note the following fact: when [G : G + ] < ∞, clearly every irreducible nontrivial unitary representation of G + appears as a subrepresentation of the representation of G obtained from it by induction.

24 CHAPTER 3 To decribe the property, let us first note that a one-parameter family G t gives rise to the gauge |·| : G → [t0 , ∞) defined by |g| = inf {s ≥ t0 ; g ∈ G s }. If the family G t satisfies the condition ∩r >t G r = G t for every t ≥ t0 , then conversely the family G t is determined by the gauge, namely, G t = {g ∈ G ; |g| ≤ t} for t ≥ t0 . Note that admissibility implies that ∩r >t G r can differ from G t only by a set of measure zero if t ≥ t0 . Clearly, the resulting family is still admissible, and so we can and will assume from now on that the family G t is indeed determined by its gauge.

Coarse admissibility implies volume growth. When G is compactly generated, coarse admissiblity for an increasing family of bounded Borel subsets G t , t > 0, of G implies that for any bounded symmetric generating set S of G, there exist a = a(S) > 0 and b = b(S) ≥ 0 such that S n ⊂ G an+b . Proof. Let S be a compact symmetric generating set. 3), we conclude that G t0 +c contains an open neighborhood of the identity. Then, assuming without loss of generality that e ∈ S, we have S ⊂ SG t0 +c S ⊂ G t1 .

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