arXiv subjects
Publications and source records attributed to Joaquin Brum.
We show that certain orderable groups admit no isolated left orders. The groups we consider are cyclic amalgamations of a free group with a general orderable group, the HNN extensions of free groups over cyclic subgroups, and a particular class of one-relator groups. In order to prove the results about orders, we develop perturbation techniques for actions of these groups on the line.
In this work we exhibit flexibility phenomena for some (countable) groups acting by order preserving homeomorphisms of the line. More precisely, we show that if a left orderable group admits an amalgam decomposition of the form $G=\mathbb{F}_n*_{\mathbb Z} \mathbb{F}_m$ where $n+m\geq 3$, then every faithful action of $G$ on the line by order preserving homeomorphisms can be approximated by another action (without global fixed points) that is not semi-conjugated to the initial action. We deduce that $\mathcal{LO}(G)$, the space of left orders of $G$, is a Cantor set. In the special case where $G=π_1(Σ)$ is the fundamental group of a closed hyperbolic surface, we found finer techniques of perturbation. For instance, we exhibit a single representation whose conjugacy class in dense in the space of representations. This entails that the space of representations without global fixed points of $π_1(Σ)$ into $Homeo_+(\mathbb R)$ is connected, and also that the natural conjugation action of $π_1(Σ)$ on $\mathcal{LO}(π_1(Σ))$ has a dense orbit.
Let $f\colon M\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus.