On the dynamics of (left) orderable groups

By Andrés Navas. 2007

131 citations recorded by the source corpus.

We study left orderable groups by using dynamical methods. We apply these techniques to study the space of orderings of these groups. We show for instance that for the case of (non-Abelian) free groups, this space is homeomorphic to the Cantor set. We also study the case of braid groups (for which the space of orderings has isolated points but contains homeomorphic copies of the Cantor set). To do this we introduce the notion of the Conradian soul of an order as the maximal subgroup which is convex and restricted to which the original ordering satisfies the so called conradian property, and we elaborate on this notion.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d81e3d8f188c1edaf80dcbb61895bfe871894e1a

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In Computational Theory and Mathematics

In Geometry and Topology

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Citations in the corpus

Foundations of Garside Theory (2013, methodology): “Navas [183] that the Dehornoy ordering is a limit of its conjugates, both in th e case of3-strand braids and in the general case using the argument of Exercise 112.”
Ordered Groups and Topology (2015, background): “That the space LO(Fn) admits no isolated points has, to date, been proved in many different ways [20, 87, 76, 58], although it was first proved by McCleary [69].”
Rigidity for $$C^1$$C1 actions on the interval arising from hyperbolicity I: solvable groups (2013, background): “Every such group admits a faithful action by homeomorphisms of the interval provided it is countable; see [29].”
GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “In an opposite direction, for some classes of groups G, it is known that no left order is isolated in LO(G): solvable groups with infinitely many left orders [122], free groups [64, 78, 99, 108, 121], free products of groups [121], and surface groups [1].”