arXiv subjects

Michael Handel

Publications and source records attributed to Michael Handel.

At least 19 recordsLinked to original sources

arXiv2026

Relative free splitting and free factor complexes: An overview

For any group $\Gamma$ and any free factor system $\mathscr A$ of $\Gamma$, the relative outer automorphism group $\text{Out}(\Gamma;\mathscr A)$ acts naturally on the relative free splitting complex $\mathcal{F\!S}(\Gamma;\mathscr A)$ and on the complex of relative free factor systems $\mathcal{F\!F}(\Gamma;\mathscr A)$, generalizing the well known actions of $\text{Out}(F_n)$ on the absolute free splitting complex $\mathcal{F\!S}(F_n)$ and the absolute free factor complex ${\mathcal{F}}(F_n)$ of the rank $n$ free group $F_n$. This overview summarizes a three part work regarding the large scale geometry of $\mathcal{F\!S}(\Gamma;\mathscr A)$ and $\mathcal{F\!F}(\Gamma;\mathscr A)$ and the geometric dynamics of the actions on these complexes by elements of $\text{Out}(\Gamma;\mathscr A)$. In Part I arXiv:1407.3508 we prove hyperbolicity of $\mathcal{F\!S}(\Gamma;\mathscr A)$ and of $\mathcal{F\!F}(\Gamma;\mathscr A)$. In Parts II and III arXiv:2212.09907, arXiv:2503.07532 we study the relation between the geometric dynamics of an element of $\text{Out}(\Gamma;\mathscr A)$ and the dynamics of its relative train track representatives. The main tool in Part II is the Two Over All Theorem, expressing an exponential flaring property of Stallings fold paths in $\mathcal{F\!S}(\Gamma;\mathscr A)$. The main tools in Part III are "filling paths", used to formulate and prove a strong version of the Two Over All Theorem.

math.GR↗

arXiv2025

Relative Free Splitting Complexes III: Stable Translation Lengths and Filling Paths

This is the last of a three part work about relative free splitting complexes $\mathcal{FS}(\Gamma,\mathscr{A})$ and their actions by relative outer automorphism groups $\text{Out}(\Gamma;\mathscr{A})$. We obtain quantitative relations between the stable translation length $\tau_\phi$ and the relative train track dynamics of~$\phi \in \Out(\Gamma;\A)$. First, if $\phi$ has an orbit with diameter bounded below by a certain constant $\Omega(\Gamma;\mathscr{A}) \ge 1$ then $\phi$ has a filling attracting lamination. Also, there is a positive lower bound $\tau_\phi \ge A(\Gamma;\mathscr{A}) > 0$ amongst all $\phi$ which have a filling attracting lamination. Both proofs rely on a study of \emph{filling paths} in a free splitting. These results are all new even for $\text{Out}(F_n)$.

math.GR↗

arXiv2022

Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem

This is the second of a three part study of relative free splitting complexes $\mathcal{FS}(\Gamma;\mathscr A)$, known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths $\tau_\phi \ge 0$ of the simplicial isometries of $\mathcal{FS}(\Gamma;\mathscr A)$ induced by relative outer automorphisms $\phi \in \text{Out}(\Gamma;\mathscr A)$, stating and proving quantitative generalizations of earlier theorems for $\text{Out}(F_n)$. The main technical result proved here in Part~II is the \emph{Two Over All Theorem}, which expresses a uniform exponential flaring property along arbitrary Stallings fold paths in $\mathcal{FS}(\Gamma;\mathscr A)$, a new result even for $\text{Out}(F_n)$. We give two applications of this theorem. First, the natural map from the relative outer space ${\mathscr O}(\Gamma;\mathscr A)$ to the relative free splitting complex $\mathcal{FS}(\Gamma;\mathscr A)$ is coarsely Lipschitz, with respect to the log-Lipschitz semimetric on~${\mathscr O}(\Gamma;\mathscr A)$. Second, if $\phi \in \text{Out}(\Gamma;\mathscr A)$ has a filling attracting lamination with expansion factor $\lambda>1$ then the stable translation length of $\phi$ acting on $\mathcal{FS}(\Gamma;\mathscr A)$ has an upper bound of the form~$B \log(\lambda)$.

math.GR↗

arXiv2020

A McCool Whitehead type theorem for finitely generated subgroups of $\mathsf{Out}(F_n)$

S. Gersten announced an algorithm that takes as input two finite sequences $\vec K=(K_1,\dots, K_N)$ and $\vec K'=(K_1',\dots, K_N')$ of conjugacy classes of finitely generated subgroups of $F_n$ and outputs: (1) $\mathsf{YES}$ or $\mathsf{NO}$ depending on whether or not there is an element $\theta\in \mathsf{Out}(F_n)$ such that $\theta(\vec K)=\vec K'$ together with one such $\theta$ if it exists and (2) a finite presentation for the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$. S. Kalajd\v{z}ievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler-Vogtmann's Outer space. New results include that the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$ is of type $\mathsf{VF}$, an equivariant version of these results, an application, and a unified approach to such questions.

math.GR↗

arXiv2019

The conjugacy problem for UPG elements of $Out(F_n)$

An element $\phi$ of the outer automorphism group $\Out(\f)$ of the rank $n$ free group $F_n$ is {\it polynomially growing} if the word lengths of conjugacy classes in $\f$ grow at most polynomially under iteration by $\phi$. It is {\it unipotent} if additionally its action on the first homology of $\f$ with integer coefficients is unipotent. In particular, if $\phi$ is polynomially growing and acts trivially on first homology with coefficients the integers mod 3 then $\phi$ is unipotent and also every polynomially growing element has a positive power that is unipotent. We solve the conjugacy problem in $\Out(\f)$ for the subset of unipotent elements. Specifically, there is an algorithm that decides if two such are conjugate in $\Out(\f)$.

math.GR↗

arXiv2019

Second bounded cohomology and WWPD

Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property --- weak proper discontinuity --- for studying the 2nd bounded cohomology of the group. We carry out a more general study of second bounded cohomology using a 'really' weak property discontinuity property known as WWPD that was introduced by Bestvina, Bromberg, and Fujiwara.

math.GR↗

arXiv2018

Virtually abelian subgroups of $IA_n(Z/3)$ are abelian

When studying subgroups of $Out(F_n)$, one often replaces a given subgroup $H$ with one of its finite index subgroups $H_0$ so that virtual properties of $H$ become actual properties of $H_0$. In many cases, the finite index subgroup is $H_0 = H \cap IA_n(Z/3)$. For which properties is this a good choice? Our main theorem states that being abelian is such a property. Namely, every virtually abelian subgroup of $IA_n(Z/3)$ is abelian.

math.GR↗

arXiv2017

Hyperbolic actions and 2nd bounded cohomology of subgroups of $\mathsf{Out}(F_n)$. Part II: Finite lamination subgroups

This is the second part of a two part work in which we prove that for every finitely generated subgroup $\Gamma < \mathsf{Out}(F_n)$, either $\Gamma$ is virtually abelian or its second bounded cohomology $H^2_b(\Gamma;\mathbb{R})$ contains an embedding of $\ell^1$. Here in Part II we focus on finite lamination subgroups $\Gamma$ --- meaning that the set of all attracting laminations of elements of $\Gamma$ is finite --- and on the construction of hyperbolic actions of those subgroups to which the general theory of Part I is applicable.

math.GR↗

arXiv2015

Hyperbolic actions and 2nd bounded cohomology of subgroups of $\text{Out}(F_n)$. Part I: Infinite lamination subgroups

In this two part work we prove that for every finitely generated subgroup $\Gamma < \text{Out}(F_n)$, either $\Gamma$ is virtually abelian or $H^2_b(\Gamma;\mathbb{R})$ contains an embedding of $\ell^1$. The method uses actions on hyperbolic spaces, for purposes of constructing quasimorphisms. Here in Part I, after presenting the general theory, we focus on the case of infinite lamination subgroups $\Gamma$ - those for which the set of all attracting laminations of all elements of $\Gamma$ is infinite - using actions on free splitting complexes of free groups.

math.GR↗

arXiv2014

Algorithmic constructions of relative train track maps and CTs

Every rotationless outer automorphism of a finite rank free group is represented by a particularly useful relative train track map called a CT. The main result of this paper is that the constructions of CTs can be made algorithmic. A key step in our argument is proving that it is algorithmic to check if an inclusion of one invariant free factor system in another is reduced. Several applications are included, as well as algorithmic constructions for relative train track maps in the general case.

math.GR↗

arXiv2014

Relative free splitting and free factor complexes I: Hyperbolicity

We study the large scale geometry of the relative free splitting complex and the relative free factor complex of the rank $n$ free group $F_n$, relative to the choice of a free factor system of $F_n$, proving that these complexes are hyperbolic. Furthermore we present the proof in a general context, obtaining hyperbolicity of the relative free splitting complex and of the relative free factor complex of a general group $\Gamma$, relative to the choice of a free factor system of $\Gamma$. The proof yields information about coarsely transitive families of quasigeodesics in each of these complexes, expressed in terms of fold paths of free splittings.

math.GR↗

arXiv2014

The free splitting complex of a free group II: Loxodromic outer automorphisms

We study the loxodromic elements for the action of $Out(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. We prove that each outer automorphism is either loxodromic, or has bounded orbits without any periodic point, or has a periodic point; and we prove that all three possibilities can occur. We also prove that two loxodromic elements are either co-axial or independent, meaning that their attracting/repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each of the alternatives in these results is also characterized in terms of the attracting/repelling lamination pairs of an outer automorphism. As an application, each attracting lamination determines its corresponding repelling lamination independent of the outer automorphism. As part of this study we describe the structure of the subgroup of $Out(F_n)$ that stabilizes the fixed point pair of a given loxodromic outer automorphism, and we give examples which show that this subgroup need not be virtually cyclic. As an application, the action of $Out(F_n)$ on the free splitting complex is not acylindrical, and its loxodromic elements do not all satisfy the WPD property of Bestvina and Fujiwara.

math.GR↗

arXiv2013

Subgroup decomposition in $\text{Out}(F_n)$, Part IV: Relatively irreducible subgroups

This is the fourth and last in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of $\text{Out}(F_n)$. In this paper we develop general ping-pong techniques for the action of $\text{Out}(F_n)$ on the space of lines of $F_n$. Using these techniques we prove the main results stated in the research announcement, Theorem C and its special case Theorem I, the latter of which says that for any finitely generated subgroup $\mathcal H$ of $\text{Out}(F_n)$ that acts trivially on homology with $\mathbb{Z}/3$ coefficients, and for any free factor system $\mathcal F$ that does not consist of (the conjugacy classes of) a complementary pair of free factors of $F_n$ nor of a rank $n-1$ free factor, if $\mathcal H$ is fully irreducible relative to $\mathcal F$ then $\mathcal H$ has an element that is fully irreducible relative to $\mathcal F$. We also prove Theorem J which, under the additional hypothesis that $\mathcal H$ is geometric relative to $\mathcal F$, describes a strong relation between $\mathcal H$ and a mapping class group of a surface. v3 and 4: Strengthened statements of the main theorems, highlighting the role of the finite generation hypothesis, and providing an alternative hypothesis. Strengthened proofs of lamination ping-pong, and a strengthened conclusion in Theorem J, for further applications.

math.GR↗

arXiv2013

Subgroup decomposition in $\text{Out}(F_n)$, Part III: Weak attraction theory

This is the third in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of $\text{Out}(F_n)$. In this paper, given an outer automorphism of $F_n$ and an attracting-repelling lamination pair, we study which lines and conjugacy classes in $F_n$ are weakly attracted to that lamination pair under forward and backward iteration respectively. For conjugacy classes, we prove Theorem F from the research annoucement, which exhibits a unique vertex group system called the "nonattracting subgroup system" having the property that the conjugacy classes it carries are characterized as those which are not weakly attracted to the attracting lamination under forward iteration, and also as those which are not weakly attracted to the repelling lamination under backward iteration. For lines in general, we prove Theorem G that characterizes exactly which lines are weakly attracted to the attracting lamination under forward iteration and which to the repelling lamination under backward iteration. We also prove Theorem H which gives a uniform version of weak attraction of lines. v3: Contains a stronger proof of Lemma 2.19 (part of the proof of Theorem G) for purposes of further applications.

math.GR↗

arXiv2013

Subgroup decomposition in Out(F_n): Introduction and Research Announcement

This is the introduction to a series of four papers that develop a decomposition theory for subgroups of Out(F_n) which generalizes the theory for elements of Out(F_n) found in the work of Bestvina, Feighn, and Handel on the Tits alternative, and which is analogous to the decomposition theory for subgroups of mapping class groups found in work of Ivanov. In this introduction we state the main theorems and we outline the contents of the whole series.

math.GR↗

arXiv2013

Subgroup decomposition in Out(F_n), Part I: Geometric Models

This is the first in a series of four papers (with research announcement posted on this arXiv) that together develop a decomposition theory for subgroups of Out(F_n). In this paper we develop further the theory of geometric EG strata of relative train track maps originally introduced in the work of Bestvina, Feighn, and Handel on the Tits alternative, with our focus trained on certain 2-dimensional models of such strata called "geometric models" and on the interesting properties of these models. A secondary purpose of this paper is to serve as a central reference for the whole series regarding basic facts of the general theory for elements of Out(F_n).

math.GR↗

arXiv2013

Subgroup decomposition in Out(F_n), Part II: A relative Kolchin theorem

This is the second in a series of four papers (with research announcement posted on this arXiv) that together develop a decomposition theory for subgroups of Out(F_n). In this paper we relativize the "Kolchin-type theorem" from the work of Bestvina, Feighn, and Handel on the Tits alternative, which describes a decomposition theory for subgroups H of Out(F_n) all of whose elements have polynomial growth. The Relative Kolchin Theorem allows subgroups H whose elements have exponential growth, as long as all such exponential growth is cordoned off in some free factor system F which is invariant under every element of H. The conclusion is that a certain finite index subgroup of H has an invariant filtration by free factor systems going from F up to the full free factor system by individual steps each of which is a "one-edge extension". We also study the kernel of the action of Out(F_n) on homology with Z/3 coefficients, and we prove Theorem B from the research announcement, which describes strong finite permutation behavior of all elements of this kernel.

math.GR↗

arXiv2012

Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of $S^2$

We show that if $M$ is a compact oriented surface of genus 0 and $G$ is a subgroup of $\Symp^\omega_\mu(M)$ which has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f \in \Symp^\omega_\mu(M)$ is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of $\Symp^\omega_\mu(M)$ then $G$ is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of $\Symp^\omega_\mu(S^2).$

math.DS↗