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Ilya Kapovich

Publications and source records attributed to Ilya Kapovich.

At least 19 recordsLinked to original sources
arXiv2026

Small Cancellation Stability and Isomorphism Rigidity for Generic Finitely Presented Groups

Let $F_m=F(a_1,\dots,a_m)$ with $m\ge 2$, and fix $q\ge 1$. For every fixed $0<λ<1$, we prove that a $q$-tuple $\mathbf W_n$ of independent uniformly random cyclically reduced words of length $n$ is \emph{$λ$-stable} with probability converging to $1$ exponentially fast. Namely, for every $Φ\in Aut(F_m)$, the tuple $Φ(\mathbf W_n)$, after cyclic reduction and symmetrization, satisfies the $C'(λ)$ small cancellation condition. Combining generic $λ$-stability with Greendlinger normal-closure rigidity and with previous results of Kapovich-Schupp-Shpilrain on generic Nielsen uniqueness and generic Whitehead rigidity we establish, for any fixed $m\ge 2, q\ge 1$, isomorphism rigidity for generic $m$-generator $q$-relator groups. Thus we show that two such generic groups $\langle a_1,\dots, a_m| r_1,\dots, r_q\rangle$ and $\langle a_1,\dots, a_m| s_1,\dots, s_q\rangle$ are isomorphic if and only if, after possibly permuting and inverting the generators $a_1,\dots, a_m$, the relator tuples $(r_1,\dots, r_q)$ and $(s_1,\dots, s_q)$ are the same, up to reordering, cyclic permutations and inverting the relators. Among the applications, we obtain a quadratic-time algorithm that generically solves the isomorphism problem for $m$-generator $q$-relator groups, and show that the number of isomorphism types represented by $m$-generator $q$-relator presentations with cyclically reduced relators of length $n$ is asymptotic to \[ \frac{(2m-1)^{qn}}{2^{m+q}m!\,q!\,n^q}. \] The proof of generic $λ$-stability relies on the use of geodesic currents and on our deterministic sufficient criterion for a $q$-tuple $\mathbb W$ in $F_m$ to be $λ$-stable in terms of the components of $\mathbb W$ being sufficiently projectively close to filling currents.

math.GR↗
arXiv2026

Nielsen classes in outer automorphism groups of free groups

For every $n\ge 8$, we construct explicit families of generating pairs of $GL(n,\mathbb Z)$ and $SL(n,\mathbb Z)$ representing countably infinitely many Nielsen equivalence classes. These pairs arise as the homology images of explicit torsion generating pairs in $Out(F_n)$ and its index-two subgroup $SOut(F_n)$, yielding countably infinitely many Nielsen classes of generating pairs in both outer groups. Within each constructed outer family, all pairs become Nielsen equivalent after one stabilization, obtained by adjoining a trivial coordinate. We derive a relation-module obstruction to Nielsen equivalence of once-stabilized generating tuples. Evans's non-cancellation examples show that this obstruction is nontrivial and can distinguish Nielsen classes of once-stabilized generating tuples. We also record a quotient relation-module refinement for possible future applications.

math.GR↗
arXiv2026

Generic-case complexity of Whitehead's algorithm, revisited

In \cite{KSS06} it was shown that with respect to the simple non-backtracking random walk on the free group $F_N=F(a_1,\dots,a_N)$ the Whitehead algorithm has strongly linear time generic-case complexity and that "generic" elements of $F_N$ are "strictly minimal" in their $Out(F_N)$-orbits. Here we generalize these results, with appropriate modifications, to a much wider class of random processes generating elements of $F_N$. We introduce the notion of a ''$(M,λ, ε)$-minimal" conjugacy class $[w]$ in $F_N$, where $M\ge 1, λ>1$ and $0<ε<1$. Roughly, being $(M,λ, ε)$-minimal means that every $ϕ\in Out(F_N)$ either increases the length $||w||_A$ by a factor of at least $λ$, or distorts the length $||w||_A$ multiplicatively by a factor $ε$-close to $1$, and that the number of automorphically minimal $[u]$ in the orbit $Out(F_N)[w]$ is bounded by $M$. We then show that if a conjugacy class $[w]$ in $F_N$ is sufficiently close to a "filling" projective geodesic current $[ν]\in PCurr(F_N)$, then, after applying a single "reducing" automorphism $ψ=ψ(ν)\in Out(F_N)$ depending on $ν$ only, the element $ψ([w])$ is $(M,λ, ε)$-minimal for some uniform constants $M,λ,ε$. Consequently, for such $[w]$, Whitehead's algorithm for the automorphic equivalence problem in $F_N$ works in quadratic time on the input $([w], [w'])$ where $[w']$ is arbitrary, and in linear time if $[w']$ is also projectively close to $[ν]$. We then show that a wide class of random processes produce "random" conjugacy classes $[w_n]$ that projectively converge to some filling current in $PCurr(F_N)$. For such $[w_n]$ Whitehead's algorithm has at most quadratic generic-case complexity.

math.GR↗
arXiv2026

Sequence distortion for metric spaces

We introduce \emph{sequence distortion spectrum}, a quasi-isometry invariant recording the large-scale distance profiles of sequences indexed by $\mathbb N$ or $\mathbb Z$ in a metric space. For a rate function $f:\mathbb N\to(0,\infty)$, extended by $f(t)=0$ for integers $t\le 0$, a sequence $(p_n)$ in a metric space $X$ is \emph{$f$-distorted} if there exists an integer $C\ge 1$ such that for all $m,n$ we have $$\frac{1}{C}f(\lfloor \frac{1}{C}|n-m|-C\rfloor)\le d(p_n,p_m)\le C f(C|n-m|+C)+C.$$ This definition implies that $f(N)=O(N)$. For rate functions, realizability depends only on the ambient quasi-isometry type and the growth type of $f$. We classify the possible power rates $f(N)=N^α$ (where $0<α\le 1)$ for Euclidean spaces: in $\mathbb R$ only the linear rate $α=1$ occurs, while in $\mathbb R^k$, $k\ge2$, the realizable exponents are exactly $1/k<α\le 1$. For a geodesic $δ$-hyperbolic space $X$, no power rate $N^α$ with $0<α<1$ occurs. The hyperbolic plane also realizes the logarithmic rate. An exponential packing bound for $X$ rules out every $o(\log N)$ rate, but a proper CAT$(-1)$ surface of unbounded geometry realizes a log--log rate. In an arbitrary simplicial tree, every realizable rate is linear up to constants. Finally, we construct two pairs of proper geodesic spaces: the first has equivalent basepoint packing functions and the second equivalent uniform packing functions; both pairs have equal asymptotic dimensions and filling-function growth classes, and isometric asymptotic cones at the chosen wedge points for every common scaling sequence and ultrafilter. Yet sequence distortion distinguishes each pair, and the second pair has bounded geometry.

math.GR↗
arXiv2026

Compressed primitivity problem in free groups

For a fixed integer $r\ge 2$, we prove that the \emph{compressed primitivity problem} in the free group $F_r=F(x_1,\dots,x_r)$ is decidable in non-deterministic polynomial time. That is, for a \emph{straight-line program} $\mathcal A$ over $\{x_1,\dots,x_r\}^{\pm1}$ representing an element $g\in F_r$, the problem of deciding whether $g$ is primitive in $F_r$ belongs to $\mathsf{NP}$, with input measured by the size of $\mathcal A$. For $r=2$, we prove that this problem is decidable in deterministic polynomial time. We also show that, in every fixed rank $r\ge 2$, automorphic minimality of the conjugacy class of a compressed word in $F_r$ is decidable in deterministic polynomial time.

math.GR↗
arXiv2026

On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups

Motivated by a classic theorem of Birman and Series about the set of complete simple geodesics on a hyperbolic surface, we study the Hausdorff dimension of the set of endpoints in $\partial F_r$ of some abstract algebraic laminations associated with free group automorphisms. For an exponentially growing outer automorphism $ϕ\in Out(F_r)$ we show that the set of endpoints $\mathcal E_{L}\subseteq \partial F_r$ of any of the \emph{attracting laminations} $L$ of $ϕ$ has Hausdorff and packing dimension $0$ for any visual metric on the boundary $\partial F_r$. Similarly that $L\subseteq \partial^2 F_r$ (where $\partial^2 F_r$ is equipped with the product metric of a visual metric) has Hausdorff dimension $0$ and packing dimension $0$. If $ϕ\in Out(F_r)$ is an atoroidal and fully irreducible, we deduce the same conclusion for the set of endpoints of the ending lamination $Λ_ϕ$ of $ϕ$ that gets collapsed by the Cannon-Thurston map $\partial F_r\to \partial G_ϕ$ for the associated free-by-cyclic group $G_ϕ=F_r\rtimes_ϕ\mathbb Z$. By contrast, the set of endpoints of any of these laminations has upper box dimension $>0$ for any visual metric on $\partial F_r$.

math.GR↗
arXiv2026

On Dehn functions for infinite group presentations

We study the behavior of Dehn functions of finitely presentable groups for presentations with finite generating sets and possibly infinite sets of defining relators. For the free abelian group $\mathbb Z^2$ of rank two on generators $a,b$, we prove that the infinite presentation $\langle a,b \mid [a^{2^k},b],\ k=0,1,2,\ldots\rangle$ has Dehn function of order $n\log n$. We also prove that, for every $0<α<2$, the group $\mathbb Z^2$ admits an infinite presentation on the same two generators whose Dehn function satisfies a global upper bound $δ(n) \le C n^α+ C$ and has matching $n_i^α$-order lower-bound peaks along an infinite sequence of lengths $n_i$. We obtain a similar result, for all $0<α<1$ for torsion-free groups $G$ admitting a finite $C'(1/6)$ small cancellation presentation on the given generators $X$. We also show that the same conclusion holds for an arbitrary finitely generated group $G$ and for some finite generating set $X$ of $G$ and for all $0<α<1$. In particular, these produce continuum many distinct growth types of Dehn functions for presentations of $\mathbb Z^2$ on the standard generators $a,b$.

math.GR↗
arXiv2026

On quantitative aspects of trace polynomials

By the classic results of Fricke and Klein, for every word $w$ in the free group $F(a,b)$ there exists a unique integer \it{trace polynomial} $f_w(x,y,z)\in Z[x,y,z]$ such that $Tr(w(A,B))=f_w(Tr A,Tr B,Tr AB)$. for all $A,B\in SL(2,C)$. We study quantitative aspects of trace polynomials. We prove an exact formula for the leading homogeneous part of $f_w$ for every nontrivial cyclically reduced word $w\in F(a,b)$. In particular, if $w=u_1\cdots u_n$ is cyclically reduced over $\{a,a^{-1},b,b^{-1}\}$, and if $N_{rs}(w)$ is the number of cyclic occurrences of $rs$, then $deg f_w=n-N_{ab}(w)-N_{b^{-1}a^{-1}}(w)=n-\frac{1}{2}(N_{ab}(w)+N_{ba}(w)+N_{a^{-1}b^{-1}}(w)+N_{b^{-1}a^{-1}}(w)).$ We obtain sharp general bounds $\lceil n/2\rceil\le deg f_w\le n$ for $w\in F(a,b)$ with cyclically reduced length $n$. We also study $deg f_w$ for random positive words and for random freely reduced and random cyclically reduced words. We obtain explicit exponential upper bounds for the growth of the $\ell_1$ and $\ell_\infty$ norms of $f_w$ and exhibit examples with exponential coefficient growth at rate $φ^n$, where $φ$ is the golden ratio. We show that for random freely reduced, random cyclically reduced and random positive words $w_n$ of length $n$ in $F(a,b)$, the size of $supp(f_{w_n})$ grows at least quadratically in $n$ and the total bit-size of $f_{w_n}$ grows at least as $cn^3$. Hence, any algorithm computing $f_w$ in totally expanded form has worst-case time complexity as well as generic-case time complexity for the above models bounded below by $Ω(n^3)$. We also give a deterministic algorithm which computes the fully expanded polynomial $f_w$ in time $O(n^5)$ and space $O(n^4)$, in terms of the input word length $n$.

math.GR↗
arXiv2025

On two-generator subgroups of mapping torus groups

We prove that if $G_ϕ=\langle F, t| t x t^{-1} =ϕ(x), x\in F\rangle$ is the mapping torus group of an injective endomorphism $ϕ: F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_ϕ$ is either free or a (finitary) sub-mapping torus. As an application we show that if $ϕ\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible atoroidal automorphism then every two-generator subgroup of $G_ϕ$ is either free or has finite index in $G_ϕ$.

math.GR↗
arXiv2024

Counting conjugacy classes of fully irreducibles: double exponential growth

Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a fixed closed surface, we consider a similar question in the $Out(F_r)$ setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilitations have natural logarithm $\le L$. Let $\mathfrak N_r(L)$ denote the number of $Out(F_r)$-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is $\le L$. We prove for $r\ge 3$ that as $L\to\infty$, the number $\mathfrak N_r(L)$ has double exponential (in $L$) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems.

math.GR↗
arXiv2023

Ascending chain condition in generic groups

We prove that for any fixed integers $m\ge 2, t\ge 1, k\ge 2$ a generic $m$-generator $t$-relator group satisfies the Ascending Chain Condition for $k$-generated subgroups.

math.GR↗
arXiv2022

Primitivity index bounds in free groups, and the second Chebyshev function

Motivated by results about "untangling" closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, $d_{prim}(g;F_N)$ and $d_{simp}(g;F_N)$, where $1\ne g\in F_N$, and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence $d_{prim}(a^nb^n; F(a,b))$ as $n\to\infty$. Answering a question of Kapovich, we prove that this sequence is unbounded and that for $n_i=lcm(1,2,\dots,i)$, we have $|d_{prim}(a^{n_i}b^{n_i}; F(a,b))-\log(n_i)|\le o(\log(n_i))$. By contrast, we show that for all $n\ge 2$, one has $d_{simp}(a^nb^n; F(a,b))=2$. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.

math.GR↗
arXiv2021

Primitivity rank for random elements in free groups

For a free group $F_r$ of finite rank $r\ge 2$ and a nontrivial element $w\in F_r$ the \emph{primitivity rank} $π(w)$ is the smallest rank of a subgroup $H\le F_r$ such that $w\in H$ and that $w$ is not primitive in $H$ (if no such $H$ exists, one puts $π(w)=\infty$). The set of all subgroups of $F_r$ of rank $π(w)$ containing $w$ as a non-primitive element is denoted $Crit(w)$. These notions were introduced by Puder in \cite{Pu14}. We prove that there exists an exponentially generic subset $V\subseteq F_r$ such that for every $w\in V$ we have $π(w)=r$ and $Crit(w)=\{F_r\}$.

math.GR↗
arXiv2021

Random trees in the boundary of Outer space

We prove that for the harmonic measure associated to a random walk on Out$(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This answers a question of M. Bestvina.

math.GT↗
arXiv2017

The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces. With an appendix by Khalid Bou-Rabee

Motivated by the results of Scott and Patel about "untangling" closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou-Rabee connects the primitivity index function to the residual finiteness growth function for $F_N$.

math.GR↗
arXiv2017

Endomorphisms, train track maps, and fully irreducible monodromies

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

math.GR↗
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↗