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Vasily A. Dolgushev

Publications and source records attributed to Vasily A. Dolgushev.

10 recordsLinked to original sources
arXiv2026

Accessing non-abelian quotients of the Grothendieck-Teichmueller group via elementary tools

Many challenging questions about the Grothendieck-Teichmueller group, $GT$, are motivated by the fact that this group receives the injective homomorphism (called the Ihara embedding) from the absolute Galois group, $G_Q$, of rational numbers. Although the question about the surjectivity of the Ihara embedding is a very challenging problem, in this paper, we construct a family of finite non-abelian quotients of $GT$ that receive surjective homomorphisms from $G_Q$. We also assemble these finite quotients into an infinite (non-abelian) profinite quotient of $GT$. We prove that the natural homomorphism from $G_Q$ to the resulting profinite group is also surjective. We give an explicit description of this profinite group. To achieve these goals, we used the groupoid $GTSh$ of $GT$-shadows for the gentle version of the Grothendieck-Teichmueller group. This groupoid was introduced in the recent paper by the second author and J. Guynee and the set $Ob(GTSh)$ of objects of $GTSh$ is a poset of certain finite index normal subgroups of the Artin braid group on 3 strands. We introduce a sub-poset $Dih$ of $Ob(GTSh)$ related to the family of dihedral groups and call it the dihedral poset. We show that each element $K$ of $Dih$ is the only object of its connected component in $GTSh$. Using the surjectivity of the cyclotomic character, we prove that, if the order of the dihedral group corresponding to $K$ is a power of 2, then the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective. We introduce the Lochak-Schneps conditions on morphisms of $GTSh$ and prove that each morphism of $GTSh$ with the target $K$ in $Dih$ satisfies the Lochak-Schneps conditions. Finally, we conjecture that the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective for every object $K$ of the dihedral poset.

math.GR↗
arXiv2024

The Action of GT-Shadows on Child's Drawings

GT-shadows are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group $\widehat{GT}$ introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB_4, that are normal in B_4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of $\widehat{GT}$. We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, $\widehat{GT}$, and the absolute Galois group G_Q of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over rationals. Finally, we describe selected examples of non-Abelian child's drawings.

math.AT↗
arXiv2024

GT-shadows for the gentle version of the Grothendieck-Teichmueller group

Let $B_3$ be the Artin braid group on 3 strands and $PB_3$ be the corresponding pure braid group. In this paper, we construct the groupoid $GTSh$ of GT-shadows for a (possibly more tractable) version $GT_0$ of the Grothendieck-Teichmueller group $GT$ introduced by D. Harbater and L. Schneps in 2000. We call this group the gentle version of $GT$ and denote it by $GT_{gen}$. The objects of $GTSh$ are finite index normal subgroups $N$ of $B_3$ satisfying the condition $N \subset PB_3$. Morphisms of $GTSh$ are called GT-shadows and they may be thought of as approximations to elements of $GT_{gen}$. We show how GT-shadows can be obtained from elements of $GT_{gen}$ and prove that $GT_{gen}$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $GTSh$. Using this result, we get a criterion for identifying genuine GT-shadows.

math.GR↗
arXiv2021

What are GT-shadows?

Let $B_4$ (resp. $PB_4$) be the braid group (resp. the pure braid group) on 4 strands and $NFI_{PB_4}(B_4)$ be the poset whose objects are finite index normal subgroups of $B_4$ that are contained in $PB_4$. In this paper, we introduce GT-shadows which may be thought of as "approximations" to elements of the profinite version $\widehat{GT}$ of the Grothendieck-Teichmueller group (see V. Drinfeld, Algebra i Analiz, 1990). We prove that GT-shadows form a groupoid whose objects are elements of $NFI_{PB_4}(B_4)$. We show that GT-shadows coming from elements of $\widehat{GT}$ satisfy various additional properties and we investigate these properties. We establish an explicit link between GT-shadows and the group $\widehat{GT}$ (see Theorem 3.8). We also present selected results of computer experiments on GT-shadows. In the appendix of this paper, we give a complete description of GT-shadows in the Abelian setting. We also prove that, in the Abelian setting, every GT-shadow comes from an element of $\widehat{GT}$. Objects very similar to GT-shadows were introduced in a paper by D. Harbater and L. Schneps in 1997. A variation of the concept of $GT$-shadows for the coarse version of $\widehat{GT}$ was studied in papers by P. Guillot.

math.AT↗
arXiv2017

The cohomology of the full directed graph complex

In his seminal paper "Formality conjecture", M. Kontsevich introduced a graph complex $GC_{1ve}$ closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the cohomology of the full directed graph complex explicitly in terms of the cohomology of $GC_{1ve}$. Applications of our results include a recent work by the first author which completely characterizes homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the stable setting.

math.KT↗
arXiv2016

On an enhancement of the category of shifted L-infinity algebras

We construct a symmetric monoidal category $LIE^{MC}$ whose objects are shifted L-infinity algebras equipped with a complete descending filtration. Morphisms of this category are "enhanced" infinity morphisms between shifted L-infinity algebras. We prove that any category enriched over $LIE^{MC}$ can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L-infinity algebra point of view. We use this construction in a subsequent paper to produce a simplicial model of a $(\infty,1)$-category whose objects are homotopy algebras of a fixed type.

math.CT↗
arXiv2015

A Version of the Goldman-Millson Theorem for Filtered L-infinity Algebras

In this paper we consider $L_{\infty}$-algebras equipped with complete descending filtrations. We prove that, under some mild conditions, an $L_{\infty}$ quasi-isomorphism $U: L \to \tilde{L}$ induces a weak equivalence between the Deligne-Getzler-Hinich (DGH) $\infty$-groupoids corresponding to $L$ and $\tilde{L}$, respectively. This paper may be considered as a modest addition to foundational paper arXiv:math/0404003 by Ezra Getzler.

math.AT↗
arXiv2015

What do homotopy algebras form?

In paper arXiv:1406.1744, we constructed a symmetric monoidal category $LIE^{MC}$ whose objects are shifted (and filtered) L-infinity algebras. Here, we fix a cooperad $C$ and show that algebras over the operad $Cobar(C)$ naturally form a category enriched over $LIE^{MC}$. Following arXiv:1406.1744, we "integrate" this $LIE^{MC}$-enriched category to a simplicial category $HoAlg^Δ_C$ whose mapping spaces are Kan complexes. The simplicial category $HoAlg^Δ_C$ gives us a particularly nice model of an $(\infty,1)$-category of $Cobar(C)$-algebras. We show that the homotopy category of $HoAlg^Δ_C$ is the localization of the category of $Cobar(C)$-algebras and infinity morphisms with respect to infinity quasi-isomorphisms. Finally, we show that the Homotopy Transfer Theorem is a simple consequence of the Goldman-Millson theorem.

math.CT↗
arXiv2012

Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction

We give a detailed proof of T. Willwacher's theorem arXiv:1009.1654 which links the cohomology of the full graph complex fGC to the cohomology of the deformation complex of the operad GER, governing Gerstenhaber algebras. We also present various prerequisites required for understanding the material of arXiv:1009.1654. In particular, we review operads, cooperads, and the cobar construction. We give a detailed exposition of the convolution Lie algebra and its properties. We prove a useful lifting property for maps from a dg operad obtained via the cobar construction. We describe in detail Willwacher's twisting construction, and then use it to work with various operads assembled from graphs, in particular, the full graph complex and its subcomplexes. These notes are loosely based on lectures given by the first author at the Graduate and Postdoc Summer School at the Center for Mathematics at Notre Dame (May 31 - June 4, 2011).

math.KT↗