Joaquín Brum's research while affiliated with Universidad de la República de Uruguay and other places
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Publications (15)
We prove a structural result for orientation-preserving actions of finitely generated solvable groups on real intervals, considered up to semi-conjugacy. As applications we obtain new answers to a problem first considered by J. F. Plante, which asks under which conditions an action of a solvable group on a real interval is semi-conjugate to an acti...
We prove various results that, given a sufficiently rich subgroup $G$ of the group of homeomorphisms on the real line, describe the structure of the other possible actions of $G$ on the real line, and address under which conditions such actions must be semi-conjugate to the natural defining action of $G$. The main assumption is that $G$ should be l...
We show that for generic homeomorphisms homotopic to the identity in a closed and oriented surface of genus $g>1$, the rotation set is given by a union of at most $2^{5g-3}$ convex sets. Examples showing the sharpness for this asymptotic order are provided.
We give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results of Alvarez-Brum-Mart\'inez-Potrie and Blanc, complete the pan...
We study the geometry of positive cones of left-invariant total orders (left-order, for short) in finitely generated groups. We introduce the {\em Hucha property} and the {\em Prieto property} for left-orderable groups. The first one means that in any left-order the corresponding positive cone is not coarsely connected, and the second one that in a...
We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness...
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...
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 = F n ∗ Z F m where n + m ⩾ 3 , then every faithful action of G on the line by order preserving homeomorphisms can be ap...
We prove that if $\Gamma$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line.
We study the space of representations $Hom(\pi_1(\Sigma), Homeo_+(\mathbb{R}))$, where $\pi_1(\Sigma)$ is the fundamental group of a hyperbolic closed surface, and $Homeo_+(\mathbb{R})$ is the group of order-preserving homeomorphisms of the real line. We focus on the subspace $Hom_{\#}$ of the representations with no global fixed points. We show th...
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 con...
Citations
... In a more recent work [4], S. Álvarez & J. Brum describe the allowed topologies for non-generic leaves of laminations that can occur when the topological type of the generic leaf is given, showing that Condition ( ) is a necessary and sufficient condition. ...
... In previous works, the authors have given examples of groups not admitting regular left-orders [1,31]. In this paper we focus instead on constructing regular leftorders. ...
Reference: Regular left-orders on groups
... In a companion paper [2], written together with Martínez and Potrie, we treated the case of minimal hyperbolic solenoids with a simply connected generic leaf. More precisely, we constructed a minimal lamination by hyperbolic surfaces such that every non-compact surface is homeomorphic to a leaf of the lamination. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/372317390688257-1465778791236_Q64/Rafael-Potrie.jpg)
... This can be used to show that some groups, for instance braid groups (see Dubrovin and Dubrovina [37], or the monograph by Dehornoy, Dynnikov, Rolfsen, and Wiest [31]), do have locally rigid actions. However the converse to this criterion does not hold, and this approach has been more fruitful in the opposite direction, namely for showing that a group has no isolated order from flexibility of the dynamical realization (see for instance the works by Navas [90], or by Alonso, and the first and third named authors [2,3], as well as by Malicet, Mann, and the last two authors [66]). One difficulty underlying this approach is that it is usually not easy to determine when two orders in LO(G) give rise to semi-conjugate actions. ...
... Note that many finitely generated groups which belong to the class F, or even to the more restricted class F 0 , do indeed admit exotic actions falling in (3), for instance as a consequence of Proposition 1.5 (but not all groups in F 0 do, see Theorem 1.20 below). Moreover in some cases there are uncountably many non-semi-conjugate such actions (see for instance Theorem 1.18 below for the case of Thompson's group F ), and the variety and flexibility of constructions suggest that in general it is too complicated to obtain a reasonably explicit description of all semi-conjugacy classes of exotic actions (however the word explicit is crucial here, as we shall see in Theorem 1.14 that such a description exists in principle). ...
... We have the following theorem which says that all expansive homeomorphisms of orientable surfaces are finitely presented, see [10,9,12,2]. We have adapted the language of this theorem from (pseudo-)Anosov diffeomorphisms to the language of Smale spaces (Anosov diffeomorphisms) and finitely presented systems (pseudo-Anosov diffeomorphisms). ...
