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Asaf Hadari

Publications and source records attributed to Asaf Hadari.

11 recordsLinked to original sources
arXiv2020

Mapping class groups of surfaces of genus $\geq 3$ do not virtually surject to $\mathbb{Z}$

We prove a well known conjecture of Nikolai Ivanov which states that if $X$ is a surface of genus $\geq 3$ (with any number of punctures and boundary components), $\rm{Mod}(X)$ is the mapping class group of $X$, and $K < \rm{Mod}(X)$ is a finite-index subgroup, then $K$ does not virtually surject to $\mathbb{Z}$. As a corollary of this we get that $H_1(Z; \mathbb{Q}) = 0$ whenever $Z$ is a finite cover of $\mathcal{M}_{g,n}$, the moduli space of complex algebraic curves of genus $g\geq 3$ with $n$ marked points.

math.GT↗
arXiv2019

Separating subgroups of mapping class groups in homological representations

Let $Γ$ be either the mapping class group of a closed surface of genus $\geq 2$, or the automorphism group of a free group of rank $\geq 3$. Given any homological representation $ρ$ of $Γ$ corresponding to a finite cover, and any term $\mathcal{I}_k$ of the Johnson filtration, we show that $ρ(\mathcal{I}_k)$ has finite index in $ρ(\mathcal{I})$, the Torelli subgroup of $Γ$. Since $[\mathcal{I}: \mathcal{I}_k] = \infty$ for $k > 1$, this implies for instance that no such representation is faithful.

math.GT↗
arXiv2018

Non virtually solvable subgroups of mapping class groups have non virtually solvable representations

Let $Σ$ be a compact orientable surface of finite type with at least one boundary component. Let $Γ\leq \textup{Mod}(Σ)$ be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation $ρ$ of $\textup{Mod}(Σ)$ such that $ρ(Γ)$ is not virtually solvable. We then apply results of Lubotzky and Meiri to show that for any random walk on such a group the probability of landing on a power, or on an element with topological entropy $0$ both decrease exponentially in the length of the walk.

math.GT↗
arXiv2017

Homological eigenvalues of lifts of pseudo-Anosov mapping classes to finite covers

Let $Σ$ be a compact orientable surface of finite type with at least one boundary component. Let $f \in \textup{Mod}(Σ)$ be a pseudo Anosov mapping class. We prove a conjecture of McMullen by showing that there exists a finite cover $\widetildeΣ \to Σ$ and a lift $\widetilde{f}$ of $f$ such that $\wt{f}_*: H_1(\wtΣ; \mathbb{Z}) \to H_1(\wtΣ; \mathbb{Z})$ has an eigenvalue off the unit circle.

math.GT↗
arXiv2015

Homological shadows of attracting laminations

Given a free group $F_n$, a fully irreducible automorphism $f \in \aut$, and a generic element $x \in F_n$, the elements $f^k(x)$ converge in the appropriate sense to an object called an attracting lamination of $f$. When the action of $f$ on $\frac{F_n}{[F_n, F_n]}$ has finite order, we introduce a homological version of this convergence, in which the attracting object is a convex polytope with rational vertices, together with a measure supported at a point with algebraic coordinates.

math.GR↗
arXiv2015

Every Infinite order mapping class has an infinite order action on the homology of some finite cover

We prove the following well known conjecture: let $Σ$ be an oriented surface of finite type whose fundamental group is a nonabelian free group. Let $ϕ\in \textup{Mod}(Σ)$ be a an infinite order mapping class. Then there exists a finite solvable cover $\widehatΣ \to Σ$, and a lift $\widehatϕ$ of $ϕ$ such that the action of $\widehatϕ$ on $H_1(\widehatΣ, \mathbb{Z})$ has infinite order. Our main tools are the theory of homological shadows, which was previously developed by the author, and Fourier analysis

math.GT↗
arXiv2014

The spectra of polynomial equations with varying exponents

We study the dependence of solutions of equations of the form $a_0 + a_1 z^{\ell_1} + ... + a_m z^{\ell_m} = 0$, on the exponents $\ell_1, ..., \ell_m$. We apply our results to equations that appear in graph theory, the theory of 3-manifolds fibering over the circle, and the theory of free-by-cyclic groups. In particular, we provide descriptions of the spectra of the Alexander polynomial of a fibered 3-manifold, Teichmüller polynomials associated to such a manifold or to a free by cyclic group, and the family of characteristic polynomials of a fixed directed graph with varying edge lengths.

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arXiv2012

Linear representations of Aut(F_r) on the homology of representation varieties

Let G be a compact semisimple linear Lie group. We study the action of Aut(F_r) on the space H_*(G^r;\QQ). We compute the image of this representation and prove that it only depends on the rank of the Lie algebra of G. We show that the kernel of this representation is always the Torrelli subgroup IA_r of Aut(F_r).

math.GR↗
arXiv2010

Simple closed curves, word length, and nilpotent quotients of free groups

We consider the fundamental group $π$ of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of $π$ has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that shows that $π$ has infinite diameter with respect to this set. Furthermore, we give a general criterion for a finitely generated group equipped with a generating set to have this property.

math.GT↗
arXiv2010

Algebraic Entropy and the Action of Mapping Class Groups on Character Varieties

We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action.

math.DS↗