Search results for "Finitely generated group"
showing 2 items of 2 documents
Small $C^1$ actions of semidirect products on compact manifolds
2020
Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eige…
Hasse diagrams and orbit class spaces
2011
Abstract Let X be a topological space and G be a group of homeomorphisms of X. Let G ˜ be an equivalence relation on X defined by x G ˜ y if the closure of the G-orbit of x is equal to the closure of the G-orbit of y. The quotient space X / G ˜ is called the orbit class space and is endowed with the natural order inherited from the inclusion order of the closure of the classes, so that, if such a space is finite, one can associate with it a Hasse diagram. We show that the converse is also true: any finite Hasse diagram can be realized as the Hasse diagram of an orbit class space built from a dynamical system ( X , G ) where X is a compact space and G is a finitely generated group of homeomo…