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Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs.

Craig A. Jensen (2002)

Publicacions Matemàtiques

It is not known whether or not the stable rational cohomology groups H*(Aut(F∞);Q) always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields...

Strong boundedness and algebraically closed groups

Barbara Majcher-Iwanow (2007)

Commentationes Mathematicae Universitatis Carolinae

Let G be a non-trivial algebraically closed group and X be a subset of G generating G in infinitely many steps. We give a construction of a binary tree associated with ( G , X ) . Using this we show that if G is ω 1 -existentially closed then it is strongly bounded.

Strongly bounded automorphism groups

A. Ivanov (2006)

Colloquium Mathematicae

A group G is strongly bounded if every isometric action of G on a metric space has bounded orbits. We show that the automorphism groups of typical countable structures with the small index property are strongly bounded. In particular we show that this is the case when G is the automorphism group of the countable universal locally finite extension of a periodic abelian group.

Structure of geodesics in the Cayley graph of infinite Coxeter groups

Ryszard Szwarc (2003)

Colloquium Mathematicae

Let (W,S) be a Coxeter system such that no two generators in S commute. Assume that the Cayley graph of (W,S) does not contain adjacent hexagons. Then for any two vertices x and y in the Cayley graph of W and any number k ≤ d = dist(x,y) there are at most two vertices z such that dist(x,z) = k and dist(z,y) = d - k. Allowing adjacent hexagons, but assuming that no three hexagons can be adjacent to each other, we show that the number of such intermediate vertices at a given distance from x and y...

Su di un problema combinatorio in teoria dei gruppi

Mario Curzio, Patrizia Longobardi, Mercede Maj (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let G be a group and n an integer ≥ 2 . We say that G has the n -permutation property ( G ∈ P n ) if, for any elements x 1 , x 2 , ⋯ , x n in G , there exists some permutation σ of { 1 , 2 , ⋯ , n } , σ ≠ i ⁢ d . such that x 1 , x 2 , ⋯ , x n = x σ ⁢ ( 1 ) , x σ ⁢ ( 2 ) , ⋯ , x σ ⁢ ( n ) . We prouve that every group G ∈ P n is an FC-nilpotent group of class ≤ n - 1 , and that a finitely generated group has the n -permutation property (for some n ) if, and only if, it is abelian by finite. We prouve also that a group G ∈ P 3 if, and only if, its derived subgroup has order at most 2.

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