Fixating Group Actions (original) (raw)

Infinite groups with fixed point properties

We construct finitely generated groups with strong fixed point properties. Let X ac be the class of Hausdorff spaces of finite covering dimension which are mod-p acyclic for at least one prime p. We produce the first examples of infinite finitely generated groups Q with the property that for any action of Q on any X ∈ X ac , there is a global fixed point. Moreover, Q may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group P that admits no non-trivial action on any manifold in X ac. In building Q, we exhibit new families of hyperbolic groups: for each n ≥ 1 and each prime p, we construct a non-elementary hyperbolic group G n,p which has a generating set of size n + 2, any proper subset of which generates a finite p-group.

Group actions having one fixed point

Mathematische Zeitschrift, 1986

in 1983. Question. Which compact Lie groups G can act smoothly (and effectively) on a closed (oriented) manifold M" of positive dimension so that the fixed point set M G consists of precisely one point ? Comments. 1. There is no loss in presuming M is connected. The component of M containing the fixed point will be invariant under the action of G. 2. If M is oriented and connected, then the diagonal action of G on M x M, g(m,m')=(gm, gm'), also fixes precisely one point, and will automatically preserve orientation. Thus, one need have no concern about preservation of orientation.

On topological groups with an approximate fixed point property

Anais da Academia Brasileira de Ciencias

A topological group G has the Approximate Fixed Point (AFP) property on a bounded convex subset C of a locally convex space if every continuous affine action of G on C admits a net ( x i ) , x i ∈ C , such that x i - g ⁢ x i ⟶ 0 for all g ∈ G . In this work, we study the relationship between this property and amenability.

On the isometrization of groups of homeomorphisms

2021

Let G be a group of homeomorphisms of a topological space X. G is (properly) isometrizable if there exists a G-invariant (proper) gauge structure on X. G is equiregular if for every x ∈ X and every open neighborhood U of x in X there is an open neighborhood V of x in X such that cl(V) ⊂ U and every y ∈ X has an open neighborhood Ny with the property that for every g ∈ G, if g(Ny) ∩ cl(V) ≠ ∅, then g(Ny) ⊂ U. G is nearly proper if for all compact subsets A and B ⊂ X, cl(⋃ { g(A) : g ∈ G and g(A) ∩ B ≠ ∅ }) is compact. The Isometrization Theorem. If X is a Hausdorff space and G\X is a paracompact regular space, then: G is isometrizable if and only if G is equiregular. The Proper Isometrization Theorem. If X is a locally compact σ-compact Hausdorff space and G\X is a regular space, then: G is properly isometrizable if and only if G is equiregular and nearly proper. G acts properly on X if for all compact subsets A and B of X, the subset GA,B = { g ∈ G: g(A) ∩ B ≠ ∅ } is compact when G ...