arXiv subjects

Cristobal Rivas

Publications and source records attributed to Cristobal Rivas.

4 recordsLinked to original sources

arXiv2022

One some planar Baumslag-Solitar actions

Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group for $n \geq 2$. We study representations of $BS(1, n)$ on the plane by orientation preserving homeomorphisms, assuming that $a$ acts as a linear map and $b$ as a map with bounded displacement. We find that the possibilities for a faithful action depend greatly on the Jordan canonical form of the map $h$ defined by the action of $a$. In case $h$ is diagonalizable over $\mathbb R$, we shall give examples or prove rigidity theorems depending on the eigenvalues. We also show some rigidity in the cases where $h$ is elliptic or parabolic. Then we give applications to the actions of $BS(1, n)$ by homeomorphisms of the torus.

math.DS↗

arXiv2022

Planar Baumslag-Solitar actions

Following previous work, where representations of $BS(1,n)$ by planar orientation preserving homeomorphisms with linear diagonalizable conjugating element was studied, we consider the elliptic and parabolic cases. As an application, we prove that there are no faithful representations of $BS(1,n)$ by toral homeomorphisms with conjugating element the Dehn twist map.

math.DS↗

arXiv2017

Ping-pong configurations and circular orders on free groups

We discuss actions of free groups on the circle with "ping-pong" dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group $F_n$ admits an isolated circular order if and only if n is even, in stark contrast with the case for linear orders. This answers a question from (Mann, Rivas, 2016). Inspired by work of Alvarez, Barrientos, Filimonov, Kleptsyn, Malicet, Menino and Triestino, we also exhibit examples of "exotic" isolated points in the space of all circular orders on $F_2$. Analogous results are obtained for linear orders on the groups $F_n \times \mathbb{Z}$.

math.GR↗

arXiv2017

Group orderings, dynamics, and rigidity

Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; as well as an analogous relationship for spaces of left orders and actions on the line. In particular, we give a complete characterization of isolated left and circular orders in terms of strong rigidity of their induced actions of G on $S^1$ and R. As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of circular orders on free groups disproving a conjecture from Baik--Samperton, and infinitely many nonconjugate isolated points in the space of left orders on the pure braid group P_3, answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders.

math.DS↗