By Lucchhini A.

**Read Online or Download A 2-generated just-infinite profinite group which is not positively generated PDF**

**Best symmetry and group books**

**Symmetry, ornament, and modularity**

This ebook discusses the origins of decorative paintings - illustrated by means of the oldest examples, courting normally from the paleolithic and neolithic a long time, and thought of from the theory-of-symmetry perspective. due to its multidisciplinary nature, it's going to curiosity quite a lot of readers: mathematicians, artists, paintings historians, architects, psychologists, and anthropologists.

- Complex cobordism and stable homotopy groups of spheres
- Finite Groups of Mapping Classes of Surfaces
- 4th Fighter Group ''Debden Eagles''
- Group theory : birdtracks, Lie's, and exceptional groups
- Symmetries and Laplacians: introduction to harmonic analysis, group representations, and applications
- Abelian Group Theory

**Additional resources for A 2-generated just-infinite profinite group which is not positively generated**

**Example text**

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 .