Free Quotients Research Articles

On the Chow theory of Quot schemes of locally free quotients

We prove a formula for the Chow groups of Quot schemes that resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This formula simultaneously generalizes the formulas for projective bundles, Grassmannian bundles, blowups, Cayley's trick, projectivizations, flops from Springer-type resolutions, and Grassmannian-type flips and flops. We also apply the formula to study the Chow groups of (i) blowups of determinantal ideals; (ii) moduli spaces of linear series on curves; and (iii) (nested) Hilbert schemes of points on surfaces.

Derived categories of Quot schemes of locally free quotients via categorified Hall products

Derived categories of Quot schemes of locally free quotients via categorified Hall products

Free quotients of favorable Calabi-Yau manifolds

Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties.

Free quotients of fundamental groups of smooth quasi-projective varieties

AbstractWe study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.

A note on Lang's conjecture for quotients of bounded domains

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains.

Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

Abstract Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial $B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots , n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.

Stability for product groups and property (τ)

We study the notion of permutation stability (or P-stability) for countable groups. Our main result provides a wide class of non-amenable product groups which are not P-stable. This class includes the product group Σ×Λ, whenever Σ admits a non-abelian free quotient and Λ admits an infinite cyclic quotient. In particular, we obtain that the groups Fm×Zd and Fm×Fn are not P-stable, for any integers m,n≥2 and d≥1. This implies that P-stability is not closed under the direct product construction, which answers a question of Becker, Lubotzky and Thom. The proof of our main result relies on a construction of asymptotic homomorphisms from Σ×Λ to finite symmetric groups starting from sequences of finite index subgroups in Σ and Λ with and without property (τ). Our method is sufficiently robust to show that the groups covered are not even flexibly P-stable, thus giving the first such non-amenable residually finite examples.

Dehn twists and Lagrangian spherical manifolds

We study Dehn twists along Lagrangian submanifolds that are finite free quotients of spheres. We describe the induced auto-equivalences to the derived Fukaya category and explain their relations to mirror symmetry.

Quotients of mapping class groups from 𝑂𝑢𝑡(𝐹_{𝑛})

We give a short proof of Masbaum and Reid’s result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield–Thurston.

Calabi‐Yau Threefolds with Small Hodge Numbers

A list of Calabi‐Yau threefolds is presented with holonomy groups that are precisely , rather than a subgroup, with small Hodge numbers, which are understood to be those manifolds with height . With the completion of a project to compute the Hodge numbers of free quotients of complete intersection Calabi‐Yau threefolds, most of which were computed in Refs. [1–3] and the remainder in Ref. [4], many new points have been added to the tip of the Hodge plot, updating the reviews by Davies and Candelas in Refs. [1,5]. In view of this and other recent constructions of Calabi‐Yau threefolds with small height, an updated list is produced.

Magnus pairs in, and free conjugacy separability of, limit groups

There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable.

ALE Ricci-flat Kähler surfaces and weighted projective spaces

We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi–Hanson, Gibbons–Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian–Yau techniques. The proof relies on a construction of good compactifications of \(\mathbb {Q}\)-Gorenstein deformations of quotient surface singularities as log del Pezzo surfaces with only cyclic quotient singularities at infinity.

Faithful actions of the absolute Galois group on connected components of moduli spaces

We use a canonical procedure associating to an algebraic number $$a$$ first a hyperelliptic curve $$C_a$$ , and then a triangle curve $$(D_a, G_a)$$ obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group $${{\mathrm{Gal}}}(\bar{{\mathbb {Q}}} /{\mathbb {Q}})$$ acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product $$C_1 \times C_2$$ of curves of respective genera $$g_1, g_2 \ge 2$$ by the action of a finite group $$G$$ ). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of the moduli space of surfaces of general type (amending an incorrect proof in a previous arXiv version of the paper); and then, as a consequence, we obtain our main result: for each element $$\sigma \in {{\mathrm{Gal}}}(\bar{{\mathbb {Q}}} /{\mathbb {Q}})$$ , not in the conjugacy class of complex conjugation, there exists a surface of general type $$X$$ such that $$X$$ and the Galois conjugate surface $$X^{\sigma }$$ have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.

Classification of unital simple Leavitt path algebras of infinite graphs

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if LK(E) and LK(F) are simple Leavitt path algebras, then LK(E) is Morita equivalent to LK(F) if and only if K0alg(LK(E))≅K0alg(LK(F)) and the graphs E and F have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K1alg(LK(E))≅K1alg(LK(F)), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras — the only missing part is determining whether the “sign of the determinant condition” is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.

The 24-cell and Calabi-Yau threefolds with Hodge numbers (1,1)

Calabi-Yau threefolds with h^11(X)=h^21(X)=1 are constructed as free quotients of a hypersurface in the ambient toric variety defined by the 24-cell. Their fundamental groups are SL(2,3), a semidirect product of Z_3 and Z_8, and Z_3 x Q_8.

Completing the web of ℤ3‐quotients of complete intersection Calabi‐Yau manifolds

AbstractWe complete the study [1] of smooth ℤ3‐quotients of complete intersection Calabi‐Yau threefolds by discussing the six new manifolds that admit free ℤ3 actions that were discovered in [2]. These manifolds were missed in [1] and complete the web of smooth ℤ3‐quotients in a nice way. We discuss the transitions between these manifolds and include also the other manifolds of the web. This leads to the conclusion that the web of ℤ3‐free quotients of complete intersection Calabi‐Yau threefolds is connected by conifold transitions.

Effective Indices of Subgroups in Baumslag–Pride Groups with Free Quotients

Given a group with at least two more generators than relations, we give an effective estimate on the minimal index of a subgroup with a nonabelian free quotient. We show that the index is bounded by a polynomial in the length of the relator words. We also provide a lower bound on the index.

On free quotients of complete intersection Calabi-Yau manifolds

In order to find novel examples of non-simply connected Calabi-Yau threefolds, free quotients of complete intersections in products of projective spaces are classified by means of a computer search. More precisely, all automorphisms of the product of projective spaces that descend to a free action on the Calabi-Yau manifold are identified.

Detecting large groups

Let G be a finitely presented group, and let p be a prime. Then G is ‘large’ (respectively, ‘ p-large’) if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a variety of new methods for detecting whether G is large or p-large. These relate to the group's profinite and pro- p completions, to its first L 2 -Betti number and to the existence of certain finite index subgroups with ‘rapid descent’. The paper draws on new topological and geometric techniques, together with a result on error-correcting codes.

Computing equations for residually free groups

We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e. of expressing G as the maximal residually free quotient of an explicit finitely presented group).