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Mark Feighn

Publications and source records attributed to Mark Feighn.

16 recordsLinked to original sources
arXiv2022

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 $θ\in \mathsf{Out}(F_n)$ such that $θ(\vec K)=\vec K'$ together with one such $θ$ if it exists and (2) a finite presentation for the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$. S. Kalajdžievski 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↗
arXiv2017

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↗
arXiv2013

Subfactor projections

When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor projections to construct an action of Out(F_n) on a finite product of hyperbolic spaces where every automorphism with exponential growth acts with positive translation length. We also prove a version of the Bounded geodesic image theorem. In the appendix, we give a sketch of the proof of the Handel-Mosher hyperbolicity theorem for the splitting complex using (liberal) folding paths.

math.GR↗
arXiv2011

Notes on Sela's work: Limit groups and Makanin-Razborov diagrams

This is the first in a planned series of papers giving an alternate approach to Zlil Sela's work on the Tarski problems. The present paper is an exposition of work of Kharlampovich-Myasnikov and Sela giving a parametrization of Hom(G,F) where G is a finitely generated group and F is a non-abelian free group.

math.GR↗
arXiv2009

A hyperbolic Out(F_n)-complex

For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $δ$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.

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↗
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↗
arXiv2005

Counting maps from a surface to a graph

Let F be a non-abelian finite rank free group, and let H_g be the fundamental group of a surface of genus g with one boundary component represented by D_g in H_g. So, H_g is the free group and D_g is the product of commutators [a_1,b_1]...[a_g,b_g]. Given x in F, we are interested in the number num(x) of primitive, i.e. root-free, images of monomorphisms (H_g,D_g) -> (F,x). Our main result is that f(g) >= 2^g where f(g)=sup num(x), where sup is taken over all elements x in F. This answers a question of Zlil Sela that is related to his work on the Tarski problem. We also show that f is independent of F and go on to obtain similar results where the monomorphisms considered are additionally required to have minimal genus.

math.GT↗
arXiv2000

Proper actions of lattices on contractible manifolds

Every lattice H in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if H acts on a contractible manifold W and if either 1) the action is properly discontinuous, or 2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1, then dim W is not less than dim G/K.

math.GT↗
arXiv1999

Mapping tori of free group automorphisms are coherent

The mapping torus of an endomorphism Φof a group G is the HNN-extension G*_G with bonding maps the identity and Φ. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.

math.GR↗
arXiv1997

The Tits Alternative for $Out(F_n)$ II: A Kolchin Type Theorem

The proof of the Tits alternative for $Out(F_n)$ is completed. The main tool is a Kolchin type theorem, proved in this paper. It states that a finitely generated subgroup of $Out(F_n)$ consisting of unipotent automorphisms can be conjugated into an upper-triangular subgroup (this is interpreted via train-tracks).

math.GT↗