arXiv subjects
Publications and source records attributed to Michele Triestino.
It is an open problem to determine whether surface groups can embed in Thompson's group $V$. We prove that any finitely generated subgroup of Thompson's group $T$ without abelian groups of arbitrary high rank is either virtually abelian or virtually free. In particular, surface groups don't embed in $T$, answering a question of Belk and Moore.
A hyperbolic-like group is a subgroup of $\operatorname{Homeo}_+(S^1)$ such that every non-trivial element has exactly two fixed points, one attracting and one repelling. We investigate the ping-pong dynamics of hyperbolic-like groups, inspired by a conjecture of Bonatti. We show the existence of a proper ping-pong partition for any pair of non-cyclic point stabilizers. More precisely, our results explicitly provide such a ping-pong partition.
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 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 locally moving, meaning that for every open interval the subgroup of elements fixing pointwise its complement, acts on it without fixed points. We show that when $G$ is a locally moving group, every $C^1$ action of $G$ on the real line is semi-conjugate to its standard action or to a non-faithful action. The situation is much wilder when considering actions by homeomorphisms: for a large class of groups, we describe uncountably many conjugacy classes of faithful minimal actions. Next, we prove structure theorems for $C^0$ actions, based on the study of laminar actions, which are actions on the line preserving a lamination. When $G$ is a group of homeomorphisms of the line acting minimally, and with a non-trivial compactly supported element, then any faithful minimal action of $G$ on the line is either laminar or conjugate to its standard action. Moreover, when $G$ is a locally moving group with a suitable finite generation condition, for any faithful minimal laminar action there is a map from the lamination to the line, called a horograding, which is equivariant with respect to the action on the lamination and the standard one, and with some extra suitable conditions. This establishes a tight relation between all minimal actions on the line of such groups, and their standard actions. Finally, based on an analysis of the space of harmonic actions, we show that for a large class of locally moving groups, the standard action is locally rigid, in the sense that sufficiently small perturbations in the compact-open topology give semi-conjugate actions.
Let $G$ be a countable group with no finitely generated subgroup of exponential growth. We show that every action of $G$ on a countable set preserving a linear (respectively, circular) order can be realised as the restriction of some action by $C^1$ diffeomorphisms on an interval (respectively, the circle) to an invariant subset. As a consequence, every action of $G$ by homeomorphisms on a compact connected one-manifold can be made $C^1$ upon passing to a semi-conjugate action. The proof is based on a functional characterisation of groups of local subexponential growth.
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 action on the line by affine transformations. We show that this is always the case for actions by $C^1$ diffeomorphisms on closed intervals. For arbitrary actions by homeomorphisms, for which this result is no longer true (as shown by Plante), we show that a semi-conjugacy to an affine action still exists in a local sense, at the level of germs near the endpoints. Finally for a vast class of solvable groups, including all solvable linear groups, we show that the family of affine actions on the line is robust, in the sense that any action by homeomorphisms on the line which is sufficiently close to an affine action must be semi-conjugate to an affine action. This robustness fails for general solvable groups, as illustrated by a counterexample.
Given a finitely generated group $G$, the possible actions of $G$ on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space $(Y, Φ)$ naturally associated with $G$ and uniquely defined up to flow equivalence, that we call the \emph{Deroin space} of $G$. We show a realisation result: every expansive flow $(Y, Φ)$ on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions.
A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is Möbius-like if every element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general, a Möbius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily (semi-)conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$, as discovered by N. Kovačević [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Here we determine simple dynamical criteria for the existence of such a (semi-)conjugacy. We show that Möbius-like subgroups of $\mathrm{Homeo}_+(\mathbb{S}^1)$ which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of $\mathrm{PSL}(2,\mathbb{R})$. On the other hand, we provide an example of elementary subgroup of $\mathrm{Diff}^\infty_+(\mathbb{S}^1)$ satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of $\mathrm{PSL}(2,\mathbb{R})$. Finally, we show that non-elementary, non-locally discrete subgroups acting with at most $N$ fixed points are conjugate to a dense subgroup of some finite central extension of $\mathrm{PSL}(2,\mathbb{R})$.
Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group $\mathsf{Diff}^ω_+(\mathbb S^1)$ of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of $\mathsf{Diff}^ω_+(\mathbb S^1)$ acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403--453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work.
In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403-453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.
We show that the action of the group $Γ=\langle a,b,c\mid a^2=b^3=c^7=abc\rangle$ on its space of left-orders has exactly two minimal components.
For a certain class of groups of piecewise linear homeomorphisms of the interval, we prove that they admit no sufficiently regular faithful action on the line. Building on previous work of Brum, Matte Bon, Rivas, and the author [arXiv:2104.14678], the new ingredient is an observation from a recent work of Hyde and Tatch Moore [arXiv:2103.14911], which allows to reduce the problem to the case of the circle, and then apply Herman-Yoccoz theory.
In [Ann. Math. 190 (2019), 657-661], James Hyde presented the first example of non-left-orderable, finitely generated subgroup of $\mathrm{Homeo}(D,\partial D)$, the group of homeomorphisms of the disk fixing the boundary. This implies that the group $\mathrm{Homeo}(D,\partial D)$ itself is not left-orderable. We revisit the construction, and present a slightly different proof of purely dynamical flavor, avoiding direct references to properties of left-orders. Our approach allows to solve the analogue problem for actions on the circle.
To every dynamical system $(X,φ)$ over a totally disconnected compact space, we associate a left-orderable group $T(φ)$. It is defined as a group of homeomorphisms of the suspension of $(X,φ)$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple, and if it is a subshift then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system $(X,φ)$, the group $T(φ)$ does not have infinite subgroups with Kazhdan's property $(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally if $(X,φ)$ has a dense orbit then the isomorphism type of the group $T(φ)$ is a complete invariant of flow equivalence of the pair $\{φ, φ^{-1}\}$. In the appendix we describe a Polish group into which $T(φ)$ embeds densely.
Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $ψ$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $ψ^*$ 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 $π_1(T)$ 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 eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds.
We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group $Γ$ has the fixed point property FW for walls (e.g. if it has property (T)), every aperiodic action of $Γ$ by diffeomorphisms that are of class $C^r$ with countably many singularities is conjugate to an action by true diffeomorphisms of class $C^r$ on a homeomorphic (possibly non-diffeomorphic) manifold. As applications, we show that Navas's result for actions of Kazhdan groups on the circle, as well as the recent solutions to Zimmer's conjecture, generalise to aperiodic actions by diffeomorphisms with countably many singularities.
We build a faithful action of Higman's group on the line by homeomorphisms, answering a question of Yves de Cornulier. As a by-product we obtain many quasimorphisms from the Higman group into the reals. We also show that every action by $C^1$-diffeomorphisms of Higman's group on the line or the circle is trivial.
This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teresópolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems 16, 1996] and recent the work of the main author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [arXiv:1608.04995; arXiv:1610.09997]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures.
This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.