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Ilya Kapovich

Publications and source records attributed to Ilya Kapovich.

At least 37 records · Page 2Linked to original sources
arXiv2017

Counting Conjugacy Classes in $Out(F_N)$

We show that if a f.g. group $G$ has a non-elementary WPD action on a hyperbolic metric space $X$, then the number of $G$-conjugacy classes of $X$-loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$. As an application we prove that for $N\ge 3$ the number of distinct $Out(F_N)$-conjugacy classes of fully irreducibles $ϕ$ from an $R$-ball in the Cayley graph of $Out(F_N)$ with $\logλ(ϕ)$ on the order of $R$ grows exponentially in $R$.

math.GR↗
arXiv2017

Stable Strata of Geodesics in Outer Space

In this paper we propose an Outer space analogue for the principal stratum of the unit tangent bundle to the Teichmüller space $\mathcal{T}(S)$ of a closed hyperbolic surface $S$. More specifically, we focus on properties of the geodesics in Teichmüller space determined by the principal stratum. We show that the analogous Outer space "principal" periodic geodesics share certain stability properties with the principal stratum geodesics of Teichmüller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichmüller space context.

math.GR↗
arXiv2017

Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell

In \cite{Ka14} we produced an algorithm for deciding whether or not an element $ϕ\in Out(F_N)$ is an iwip ("fully irreducible") automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this paper we refine the algorithm from \cite{Ka14} by eliminating these inefficient features, and also by eliminating any use of mapping class groups algorithms. Our main result is to produce, for any fixed $N\ge 3$, an algorithm which, given a topological representative $f$ of an element $ϕ$ of $Out(F_N)$, decides in polynomial time in terms of the "size" of $f$, whether or not $ϕ$ is fully irreducible. In addition, we provide a train track criterion of being fully irreducible which covers all fully irreducible elements of $Out(F_N)$, including both atoroidal and non-atoroidal ones. We also give an algorithm, alternative to that of Turner, for finding all the indivisible Nielsen paths in an expanding train track map, and estimate the complexity of this algorithm. An appendix by Mark Bell provides a polynomial upper bound, in terms of the size of the topological representative, on the complexity of the Bestvina-Handel algorithm \cite{BH92} for finding either an irreducible train track representative or a topological reduction.

math.GR↗
arXiv2016

McMullen polynomials and Lipschitz flows for free-by-cyclic groups

Consider a group G and an epimorphism u_0:G\to\Z inducing a splitting of G as a semidirect product ker(u_0)\rtimes_φ\Z with ker(u_0) a finitely generated free group and φ\in Out(ker(u_0)) representable by an expanding irreducible train track map. Building on our earlier work [Dynamics on free-by-cyclic groups, arXiv:1301.7739], in which we we realized G as π_1(X) for an Eilenberg-Maclane 2-complex X equipped with a semiflow ψ, and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant \m for (X,ψ) and investigate its properties. Specifically, \m determines a convex polyhedral cone \C_X in H^1(G;\R), a convex, real-analytic function \H:\C_X\to\R, and specializes to give an integral Laurent polynomial \m_u(ζ) for each integral u\in\C_X. We show that \C_X is equal to the "cone of sections" of (X,ψ) (the convex hull of all cohomology classes dual to sections of of ψ), and that for each (compatible) cross section Θ_u with first return map f_u:Θ_u\toΘ_u, the specialization \m_u(ζ) encodes the characteristic polynomial of the transition matrix of f_u. More generally, for every class u\in\C_X there exists a geodesic metric d_u and a codimension-1 foliation Ω_u of X transverse to ψso that after reparametrizing the flow ψ^u_s maps leaves of Ω_u to leaves via a local e^{s\H(u)}-homothety. Among other things, we additionally prove that \C_X is equal to (the cone over) the component of the BNS-invariant containing u_0 and that each primitive integral u\in\C_X induces a splitting of G as an ascending HNN-extension over a finite-rank free group along an injective endomorphism ϕ_u. For any such splitting, we show that the stretch factor of ϕ_u is exactly given by e^{\H(u)}. In particular, we see that \C_X and \H depend only on the group G and epimorphism u_0.

math.GT↗
arXiv2016

An integral weight realization theorem for subset currents on free groups

We prove that if $N\ge 2$ and $α: F_N\to π_1(Γ)$ is a marking on $F_N$, then for any integer $r\ge 2$ and any $F_N$-invariant collection of non-negative integral "weights" associated to all subtrees $K$ of $\widetilde Γ$ of radius $\le r$ satisfying some natural "switch" conditions, there exists a finite cyclically reduced folded $Γ$-graph $Δ$ realizing these weights as numbers of "occurrences" of $K$ in $Δ$. As an application, we give a new, more direct and explicit, proof of one of the main results of our paper with Nagnibeda \cite{KN3} stating that for any $N\ge 2$ the set $\gcnr$ of all rational subset currents is dense in the space $\gcn$ of subset currents on $F_N$. We also answer one of the questions (Problem 10.11) posed in \cite{KN3}. Thus we prove that if a nonzero $μ\in \gcn$ has all weights with respect to some marking being integers, then $μ$ is the sum of finitely many "counting" currents corresponding to nontrivial finitely generated subgroups of $F_N$.

math.GR↗
arXiv2016

Conical limit points and the Cannon-Thurston map

Let $G$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $Z$ so that there exists a continuous $G$-equivariant map $i:\partial G\to Z$, which we call a \emph{Cannon-Thurston map}. We obtain two characterzations (a dynamical one and a geometric one) of conical limit points in $Z$ in terms of their pre-images under the Cannon-Thurston map $i$. As an application we prove, under the extra assumption that the action of $G$ on $Z$ has no accidental parabolics, that if the map $i$ is not injective then there exists a non-conical limit point $z\in Z$ with $|i^{-1}(z)|=1$. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if $G$ is a non-elementary torsion-free word-hyperbolic group then there exists $x\in \partial G$ such that $x$ is not a "controlled concentration point" for the action of $G$ on $\partial G$.

math.GR↗
arXiv2016

Nielsen equivalence in a class of random groups

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction.

math.GR↗
arXiv2015

Cannon-Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let $F_N$ denote a free groups of finite rank $N\ge 3$ and consider a \emph{convex cocompact} subgroup $Γ\le Out(F_N)$, i.e. one for which the orbit map from $Γ$ into the free factor complex of $F_N$ is a quasi-isometric embedding. The subgroup $Γ$ determines an extension $E_Γ$ of $F_N$, and the main theorem of Dowdall--Taylor \cite{DT1} states that in this situation $E_Γ$ is hyperbolic if and only if $Γ$ is purely atoroidal. Here, we give an explicit geometric description of the Cannon--Thurston maps $\partial F_N\to\partial E_Γ$ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon--Thurston map, showing that this map has multiplicity at most $2N$. This theorem generalizes the main result of Kapovich and Lustig \cite{KapLusCT} which treats the special case where $Γ$ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of $Γ$ to the space of laminations of the free group (with the Chabauty topology) is not continuous.

math.GR↗
arXiv2015

On purely loxodromic actions

We construct an example of an isometric action of $F(a,b)$ on a $δ$-hyperbolic graph $Y$, such that this action is acylindrical, purely loxodromic, has asymptotic translation lengths of nontrivial elements of $F(a,b)$ separated away from $0$, has quasiconvex orbits in $Y$, but such that the orbit map $F(a,b)\to Y$ is not a quasi-isometric embedding.

math.GR↗
arXiv2015

Musings on generic-case complexity

We propose a more general definition of generic-case complexity, based on using a random process for generating inputs of an algorithm and using the time needed to generate an input as a way of measuring the size of that input.

cs.CC↗
arXiv2015

A train track directed random walk on $Out(F_r)$

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element $ϕ_n\in Out(F_r)$, obtained after $n$ steps of a simple random walk on $Out(F_r)$, is fully irreducible with probability tending to 1 as $n\to\infty$. In this paper we construct a natural "train-track directed" random walk $\mathcal W$ on $Out(F_r)$ (where $r\ge 3$). We show that, for the element $ϕ_n\in Out(F_r)$, obtained after $n$ steps of this random walk, with asymptotically positive probability the element $ϕ_n$ has the following properties: $ϕ_n$ is an ageometric fully irreducible, which admits a train-track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that $ϕ_n$ has "rotationless index" $\frac{3}{2}-r$ (so that the geometric index of the attracting tree $T_{ϕ_n}$ of $ϕ_n$ is $2r-3$), has index list $\{\frac{3}{2}-r\}$ and the ideal Whitehead graph being the complete graph on $2r-1$ vertices, and that the axis bundle of $ϕ_n$ in the Outer space $CV_r$ consists of a single axis.

math.GR↗
arXiv2015

Dynamics on free-by-cyclic groups

Given a free-by-cyclic group $G = F_N \rtimes_φ\mathbb{Z}$ determined by any outer automorphism $φ\in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $ε\in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $Θ_u \subset X$ such that: (1) the inclusion $Θ_u \to X$ is $π_1$-injective and $π_1(Θ_u) = \mathrm{ker}(u)$, (2) $u(ε) = χ(Θ_u)$, (3) $Θ_u \subset X$ is a section of the semiflow and the first return map to $Θ_u$ is an expanding irreducible train track map representing $φ_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{φ_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $φ_u$ is precisely $\mathfrak H(u)$, (5) if $φ$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $φ_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.

math.GT↗
arXiv2014

Cannon-Thurston fibers for iwip automorphisms of $F_N$

For any atoroidal iwip $ϕ\in Out(F_N)$ the mapping torus group $G_ϕ=F_N\rtimes_ϕ e$ is hyperbolic, and the embedding $ι: F_N \overset{\lhd}{\longrightarrow} G_ϕ$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat ι: \partial F_N \to \partial G_ϕ$. We prove that for any $ϕ$ as above, the map $\hat ι$ is finite-to-one and that the preimage of every point of $\partial G_ϕ$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_ϕ$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ where $w\in F_N, m\ne 0$, and that there are at most $4N-5$ distinct $F_N$-orbits of such {\em singular} points in $\partial G_ϕ$ (for the translation action of $F_N$ on $\partial G_ϕ$). By contrast, we show that for $k=1,2$ there are uncountably many points $S\in \partial G_ϕ$ (and thus uncountably many $F_N$-orbits of such $S$) with exactly $k$ preimages in $\partial F_N$.

math.GR↗
arXiv2014

Unbounded asymmetry of stretch factors

A result of Handel-Mosher guarantees that the ratio of logarithms of stretch factors of any fully irreducible automorphism of the free group $F_N$ and its inverse is bounded by a constant $C_N$. In this short note we show that this constant $C_N$ cannot be chosen independent of $N$.

math.GR↗
arXiv2013

Invariant laminations for irreducible automorphisms of free groups

For every atoroidal iwip automorphism $ϕ$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(ϕ)$ is obtained as "diagonal closure" of the support of the backward limit current $μ_-(ϕ)$. This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of $T_+(ϕ)$, in terms of Bestvina-Feighn-Handel's "stable lamination" associated to $ϕ$.

math.GR↗
arXiv2013

Spectral rigidity of automorphic orbits in free groups

It is well-known that a point $T\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichmüller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not conjugate to a power of $[a,b]$. We also include an appended corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups $H\le Out(F_N)$. After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that $H$ be finitely generated. The corrigendum provides an alternative proof of Lemma~5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.

math.GR↗