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Michael Handel

Publications and source records attributed to Michael Handel.

At least 37 records · Page 2Linked to original sources
arXiv2012

Lipschitz retraction and distortion for subgroups of Out(F_n)

Given a free factor A of the rank n free group F_n, we characterize when the subgroup of Out(F_n) that stabilizes the conjugacy class of A is distorted in Out(F_n). We also prove that the image of the natural embedding of Aut(F_{n-1}) in Aut(F_n) is nondistorted, that the stabilizer in Out(F_n) of the conjugacy class of any free splitting of F_n is nondistorted, and we characterize when the stabilizer of the conjugacy class of an arbitrary free factor system of F_n is distorted. In all proofs of nondistortion, we prove the stronger statement that the subgroup in question is a Lipschitz retract. As applications we determine Dehn functions and automaticity for Out(F_n) and Aut(F_n).

math.GR↗
arXiv2012

Entropy zero area preserving diffeomorphisms of $S^2$

In this paper we formulate and prove a structure theorem for area preserving diffeomorphisms of genus zero surfaces with zero entropy. As an application we relate the existence of faithful actions of a finite index subgroup of the mapping class group of a closed surface $Σ_g$ on $S^2$ by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups $MCG(S, \partial S)$ with non-trivial first cohomology.

math.DS↗
arXiv2011

Triviality of some representations of $MCG(S_g)$ in $GL(n,C), Diff(S^2)$ and $Homeo(T^2)$

We show the triviality of representations of the mapping class group of a genus $g$ surface in $GL(n,C), Diff(S^2)$ and $Homeo(T^2)$ when appropriate restrictions on the genus $g$ and the size of $n$ hold. For example, if $S_g$ is a surface of finite type and $ϕ: MCG(S_g) \to GL(n,C)$ is a homomorphism, then $ϕ$ is trivial provided the genus $g \ge 3$ and $n < 2g$. We also show that if $S_g$ is a closed surface with genus $g \ge 7$, then every homomorphism $ϕ: MCG(S_g) \to Diff(S^2)$ is trivial and that if $g \ge 3$, then every homomorphism $ϕ: MCG(S_g) \to Homeo(T^2)$ is trivial.

math.GT↗
arXiv2009

Subgroup classification in Out(F_n)

For any subgroup H of Out(F_n), either H has a finite index subgroup that fixes the conjugacy class of some proper, nontrivial free factor of F_n, or H contains a fully irreducible element phi, meaning that no positive power of phi fixes the conjugacy class of any proper, nontrivial free factor of F_n.

math.GR↗
arXiv2009

The Recognition Theorem for Out(F_n)

Our goal is to find dynamic invariants that completely determine elements of the outer automorphism group $\Out(F_n)$ of the free group $F_n$ of rank $n$. To avoid finite order phenomena, we do this for {\it forward rotationless} elements. This is not a serious restriction. For example, there is $K_n>0$ depending only on $n$ such that, for all $ϕ\in\Out(F_n)$, $ϕ^{K_n}$ is forward rotationless. An important part of our analysis is to show that rotationless elements are represented by particularly nice relative train track maps.

math.GR↗
arXiv2008

Global fixed points for centralizers and Morita's Theorem

We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeomorphism with infinitely many global fixed points. As another application we give an elementary proof of Morita's Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of diffeormorphisms of $S$ and we improve the lower bound for $g$ from 5 to 3.

math.DS↗
arXiv2007

Complete semi-conjugacies for psuedo-Anosov homeomorphisms

Suppose $S$ is a surface of genus $\ge 2 $, $f: S \to S$ is a surface homeomorphism isotopic to a pseudo-Anosov map $α$ and suppose $\ti S$ is the universal cover of $S$ and $F$ and $A$ are lifts of $f$ and $α$ respectively. We show there is a semiconjugacy $Θ: \ti S \to \bar Ł^s \times \bar Ł^u$ from $F$ to $\bar A$, where $\bar Ł^s$ ($\bar Ł^u$) is the completion of the $R$-tree of leaves of the stable (resp. unstable) foliation for $A$ and $\bar A$ is the map induced by $A$. We also generalize a result of Markovich and show that for any $g \in Homeo(S)$ which commutes with $f$ and has identity lift $G : \ti S \to \ti S$ and for any $(c,w)$ in the image of $Θ$ each component of $Θ^{-1}(c,w)$ is $G$-invariant.

math.DS↗
arXiv2007

Commensurations of Out(F_n)

Let $\Out(F_n)$ denote the outer automorphism group of the free group $F_n$ with $n>3$. We prove that for any finite index subgroup $Γ<\Out(F_n)$, the group $\Aut(Γ)$ is isomorphic to the normalizer of $Γ$ in $\Out(F_n)$. We prove that $Γ$ is {\em co-Hopfian} : every injective homomorphism $Γ\to Γ$ is surjective. Finally, we prove that the abstract commensurator $\Comm(\Out(F_n))$ is isomorphic to $\Out(F_n)$.

math.GR↗
arXiv2006

Abelian subgroups of \Out(F_n)

We classify abelian subgroups of Out(F_n) up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element ϕinto a composition of finitely many elements and then use these elements to generate an abelian subgroup A(ϕ) that contains ϕ. The main theorem is that up to finite index every abelian subgroup is realized by this construction. As an application we classify, up to finite index, abelian subgroups of Out(F_n) and of IA with maximal rank.

math.GR↗
arXiv2006

Fixed Points of abelian actions

We prove that if $\F$ is an abelian group of $C^1$ diffeomorphisms isotopic to the identity of a closed surface $S$ of genus at least two then there is a common fixed point for all elements of $\F.$

math.DS↗
arXiv2006

Axes in Outer Space

We develop a notion of axis in the Culler--Vogtmann outer space X_r of a finite rank free group F_r, with respect to the action of a nongeometric, fully irreducible outer automorphism phi. Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmuller space, X_r has no natural metric, and phi seems not to have a single natural axis. Instead our axes for phi, while not unique, fit into an ``axis bundle'' A_phi with nice topological properties: A_phi is a closed subset of X_r proper homotopy equivalent to a line, it is invariant under phi, the two ends of A_phi limit on the repeller and attractor of the source--sink action of phi on compactified outer space, and A_phi depends naturally on the repeller and attractor. We propose various definitions for A_phi, each motivated in different ways by train track theory or by properties of axes in Teichmuller space, and we prove their equivalence.

math.GR↗
arXiv2006

The expansion factors of an outer automorphism and its inverse

A fully irreducible outer automorphism phi of the free group F_n of rank n has an expansion factor which often differs from the expansion factor of the inverse of phi. Nevertheless, we prove that the ratio between the logarithms of the expansion factors of phi and its inverse is bounded above by a constant depending only on the rank n. We also prove a more general theorem applying to an arbitrary outer automorphism of F_n and its inverse, and their entire spectrum of expansion factors.

math.GR↗
arXiv2006

Parageometric outer automorphisms of free groups

We study those fully irreducible outer automorphisms phi of a finite rank free group F_r which are ``parageometric'', meaning that the attracting fixed point of phi in the boundary of outer space is a geometric R-tree with respect to the action of F_r, but phi itself is not a geometric outer automorphism in that it is not represented by a homemorphism of a surface. Our main result shows that the expansion factor of phi is strictly larger than the expansion factor of the inverse of phi. As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism phi is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric R-trees.

math.GR↗
arXiv2005

Fixed Points of abelian actions on $S^2$

We prove that if $F$ is a finitely generated abelian group of orientation preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set then there is a common fixed point for all elements of $F.$ We also show that if $F$ is any abelian subgroup of orientation preserving $C^1$ diffeomorphisms of $S^2$ then there is a common fixed point for all elements of a subgroup of $F$ with index at most two.

math.DS↗
arXiv2005

Distortion Elements in Group actions on surfaces

If $\G$ is a finitely generated group with generators $\{g_1,...,g_j\}$ then an infinite order element $f \in \G$ is a {\em distortion element} of $\G$ provided $\displaystyle{\liminf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface and let $\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result shows that if $S$ has genus at least two and if $f$ is a distortion element in some finitely generated subgroup of $\Diff(S)_0$, then $\supp(μ) \subset \Fix(f)$ for every $f$-invariant Borel probability measure $μ$. Related results are proved for $S = T^2$ or $S^2$. For $μ$ a Borel probability measure on $S$, denote the group of $C^1$ diffeomorphisms that preserve $μ$ by $\Diff_μ(S)$. We give several applications of our main result showing that certain groups, including a large class of higher rank lattices, admit no homomorphisms to $\Diff_μ(S)$ with infinite image.

math.DS↗
arXiv2003

Periodic points of Hamiltonian surface diffeomorphisms

The main result of this paper is that every non-trivial Hamiltonian diffeomorphism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period. The same result is true for S^2 provided the diffeomorphism has at least three fixed points. In addition we show that up to isotopy relative to its fixed point set, every orientation preserving diffeomorphism F: S --> S of a closed orientable surface has a normal form. If the fixed point set is finite this is just the Thurston normal form.

math.DS↗
arXiv2003

Area preserving group actions on surfaces

Suppose G is an almost simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group. For example any finite index subgroup of SL(3,Z) is such a group. The main result of this paper is that every action of G on a closed oriented surface by area preserving diffeomorphisms factors through a finite group.

math.DS↗