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Martin Lustig

Publications and source records attributed to Martin Lustig.

At least 37 records · Page 2Linked to original sources
arXiv2011

Domains of proper discontinuity on the boundary of Outer space

Motivated by the work of McCarthy and Papadopoulos for subgroups of mapping class groups, we construct domains of proper discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any "sufficiently large" subgroup of $Out(F_N)$ (that is, a subgroup containing a hyperbolic iwip). As a corollary we prove that for $N\ge 3$ the action of $Out(F_N)$ on the subset of $\mathbb PCurr(F_N)$ consisting of all projectivized currents with full support is properly discontinuous.

math.GR↗
arXiv2011

Stabilizers of $\mathbb R$-trees with free isometric actions of $F_N$

We prove that if $T$ is an $\mathbb R$-tree with a minimal free isometric action of $F_N$, then the $Out(F_N)$-stabilizer of the projective class $[T]$ is virtually cyclic. For the special case where $T=T_+(ϕ)$ is the forward limit tree of an atoroidal iwip element $ϕ\in Out(F_N)$ this is a consequence of the results of Bestvina, Feighn and Handel, via very different methods. We also derive a new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip (not necessarily atoroidal): we prove that every such subgroup $G\le Out(F_N)$ is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of $Out(F_N)$ is due to Bestvina, Feighn and Handel.

math.GR↗
arXiv2010

Are large distance Heegaard splittings generic ?

In a previous paper we introduced a notion of "genericity" for countable sets of curves in the curve complex of a surface S, based on the Lebesgue measure on the space of projective measured laminations in S. With this definition we prove that for each fixed g > 1 the set of irreducible genus g Heegaard splittings of high distance is generic, in the set of all irreducible Heegaard splittings. Our definition of "genericity" is different and more intrinsic then the one given via random walks.

math.GT↗
arXiv2010

Ping-pong and Outer space

We prove that if $ϕ,ψ\in Out(F_N)$ are hyperbolic iwips (irreducible with irreducible powers) such that $<ϕ,ψ>\le Out(F_N)$ is not virtually cyclic then some high powers of $ϕ$ and $ψ$ generate a free subgroup of rank two, all of whose nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of $Out(F_N)$ is strongly analogous to being a pseudo-Anosov element of a mapping class group, so the above result provides analogs of "purely pseudo-Anosov" free subgroups of $Out(F_N)$.

math.GR↗
arXiv2009

Horizontal Dehn Surgery and genericity in the curve complex

We introduce a general notion of "genericity" for countable subsets of a space with Borel measure, and apply it to the set of vertices in the curve complex of a surface S, interpreted as subset of the space of projective measured laminations in S, equipped with its natural Lebesgue measure. We prove that, for any 3-manifold M, the set of curves c on a Heegaard surface S in M, such that every non-trivial Dehn twist at c yields a Heegaard splitting of high distance, is generic in the set of all essential simple closed curves on S. Our definition of "genericity" is different and more intrinsic than alternative such existing notions, given e.g. via random walks or via limits of quotients of finite sets.

math.GT↗
arXiv2009

Free group automorphisms with many fixed points at infinity

A concrete family of automorphisms alpha_n of the free group F_n is exhibited, for any n > 2, and the following properties are proved: alpha_n is irreducible with irreducible powers, has trivial fixed subgroup, and has 2n-1 attractive as well as 2n repelling fixed points at bdry F_n. As a consequence of a recent result of V Guirardel there can not be more fixed points on bdry F_n, so that this family provides the answer to a question posed by G Levitt.

math.GR↗
arXiv2009

$\R$-trees, dual laminations, and compact systems of partial isometries

Let $\FN$ be a free group of finite rank $N \geq 2$, and let $T$ be an $\R$-tree with a very small, minimal action of $\FN$ with dense orbits. For any basis $\CA$ of $\FN$ there exists a {\em heart} $K_{\CA} \subset \bar T$ (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries $a_{i} : K_{\CA} \cap a_{i} K_{\CA} \to a_{i}\inv K_{\CA} \cap K_{\CA}$, for each $a_{i} \in \CA$, defines a tree $T_{(K_{\CA}, \CA)}$ which contains an isometric copy of $T$ as minimal subtree.

math.GR↗
arXiv2009

Intersection form, laminations and currents on free groups

Let $F_N$ be a free group of rank $N\ge 2$, let $μ$ be a geodesic current on $F_N$ and let $T$ be an $\mathbb R$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $ $ is equal to zero if and only if the support of $μ$ is contained in the dual algebraic lamination $L^2(T)$ of $T$. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. As another application, we define the notion of a \emph{filling} element in $F_N$ and prove that filling elements are "nearly generic" in $F_N$. We also apply our results to the notion of \emph{bounded translation equivalence} in free groups.

math.GT↗
arXiv2009

Geometric Intersection Number and analogues of the Curve Complex for free groups

For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \emph{geometric intersection form} \[ <, >: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \] Here $\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\bar{cv}(F_N)$ consists of all \emph{very small} minimal isometric actions of $F_N$ on $\mathbb R$-trees. The projectivization of $\bar{cv}(F_N)$ provides a free group analogue of Thurston's compactification of the Teichmüller space. As an application, using the \emph{intersection graph} determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.

math.GR↗
arXiv2008

The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth

We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(α)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $α$. Furthermore, we show that $\FN \rtimes_α \Z$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(α)$. As an application, we use a result of Drutu-Sapir to deduce that $\FN \rtimes_α \Z$ has Rapic Decay.

math.GR↗
arXiv2008

High distance Heegaard splittings via fat train tracks

We define "fat" train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in the three sphere.

math.GT↗
arXiv2008

Automorphisms of free groups have asymptotically periodic dynamics

We show that every automorphism $α$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $α^q$ such that every element of the compactum $F_k \cup \partial F_k$ converges to a fixed point under iteration of $α^q$.

math.GR↗
arXiv2007

Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\R$-trees

We continue in this article the study of laminations dual to very small actions of a free group F on R-trees. We prove that this lamination determines completely the combinatorial structure of the R-tree (the so-called observers' topology). On the contrary the metric is not determined by the lamination, and an R-tree may be equipped with different metrics which have the same observers' topology.

math.GR↗
arXiv2007

$\R$-trees and laminations for free groups I: Algebraic laminations

This paper is the first of a sequence of three papers, where the concept of an $\mathbb R$-tree dual to a measured geodesic lamination in a hyperbolic surface is generalized to arbitrary $\mathbb R$-trees provided with a (very small) action of the free group $F_N$ of finite rank $N\geq 2$ by isometries. Three different definitions are given and they are proved to be equivalent. We also describe the topology and Out$(F_N)$-action on the space of laminations.

math.GR↗
arXiv2006

The actions of $Out(F_k)$ on the boundary of Outer space and on the space of currents: minimal sets and equivariant incompatibility

We prove that for $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb PCurr(F_k)$ that is either $Out(F_k)$-equivariant or $Out(F_k)$-anti-equivariant. Here $\partial CV(F_k)$ is the "length-function" boundary of Culler-Vogtmann's Outer space $CV(F_k)$, and $\mathbb PCurr(F_k)$ is the space of projectivized geodesic currents for $F_{k}$. We also prove that, if $k\ge 3$, for the action of $Out(F_k)$ on $\mathbb PCurr(F_{k})$ and for the diagonal action of $Out(F_k)$ on the product space $\partial CV(F_k)\times \mathbb PCurr(F_k)$ there exist unique non-empty minimal closed $Out(F_k)$-invariant sets. Our results imply that for $k\ge 3$ any continuous $Out(F_k)$-equivariant embedding of $CV(F_k)$ into $\mathbb PCurr(F_k)$ (such as the Patterson-Sullivan embedding) produces a new compactification of Outer space, different from the usual "length-function" compactification $\bar{CV(F_k)}=CV(F_k)\cup \partial CV(F_k)$.

math.GR↗
arXiv2002

A finiteness result for Heegaard splittings

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of $3g - 3$ disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to fact that in general there can exist infinitely many such systems of disks which satisfy just the Casson-Gordon rectangle condition.

math.GT↗