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## Embeddings of automorphism groups of free groups into automorphism groups of affine algebraic varieties

Cornell University
,
2022.
No. 2207.13072.

A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the auto\-morphism group Aut(F_n) of the free group F_n of rank n. The automorphism groups of such varieties are nonlinear and contain the braid group B_n on n strands for n > 2, and are nonamenable for n > 1. As an application, it is proved that for n\geqslant 3, every Cremona group of rank > 3n-2 contains the groups Aut(F_n) and B_n. This bound is 1 better than the one published earlier by the author; with respect to B_n the order of its growth rate is one less than that of the bound following from the paper by D. Krammer. The basis of the construction are triplets (G, R, n), where G is a connected semisimple algebraic group and R is a closed subgroup of its maximal torus.

Shafarevich A., Математический сборник 2017 Т. 208 № 2 С. 285-310

Let k be an algebraically closed field of characteristic zero and Ga = (k, +) the additive group of k. An algebraic variety X is said to be flexible if the tangent space at every regular point of X is generated by the tangent vectors to orbits of various regular actions of Ga. In 1972, ...

Added: December 5, 2018

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2022. No. 2207.08912.

Considering a certain construction of algebraic varieties X endowed with an algebraic action of the group Aut(Fn), n < ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family F of X’s such that Aut(Fn) embeds intoAut(X). For n > 3, this implies nonlinearity, and for n > 2, ...

Added: July 20, 2022

V. L. Popov, Mathematical notes 2018 Vol. 103 No. 5 P. 811-819

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan. ...

Added: April 13, 2018

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2106.02072.

For each integer n>0, we construct a series of irreducible algebraic varieties X, for which the automorphism group Aut(X) contains as a subgroup the automorphism group Aut(F_n) of a free group F_n of rank n. For n > 1, such groups Aut(X) are nonamenable, and for n > 2, they are nonlinear and contain the ...

Added: June 7, 2021

Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al., Compositio Mathematica 2011 Vol. 147 No. 2 P. 428-466

Let k be a field of characteristic zero, let G be a connected reductive algebraic group
over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces
the adjoint action of G on g. We investigate the ...

Added: March 17, 2013

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1502.02167.

According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and the s-dimensional projectice space with positive s. We show that the classical proof of this theorem actually works only in characteristic 0 and ...

Added: February 10, 2015

Р.С. Авдеев, Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3-48

For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...

Added: October 22, 2014

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2017. No. 1707.06914 [math.AG].

We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) ...

Added: July 24, 2017

Р.С. Авдеев, Математические заметки 2013 Т. 94 № 1 С. 22-35

For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G, we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups. ...

Added: February 25, 2014

Р.С. Авдеев, Труды Московского математического общества 2011 Т. 72 № 1 С. 5-62

We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugacy. ...

Added: February 25, 2014

Р.С. Авдеев, Труды Московского математического общества 2010 Т. 71 С. 235-269

A spherical homogeneous space G/H of a connected semisimple algebraic group G is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group G. All the excellent affine spherical homogeneous spaces are classified up to isomorphism. ...

Added: February 25, 2014

В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 475 № 1 С. 14-16

Даны классификации неприводимых представлений простых алгебраических групп модальностей 0, 1 и 2. ...

Added: May 3, 2017

Р.С. Авдеев, Известия РАН. Серия математическая 2010 Т. 74 № 6 С. 3-26

The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical ...

Added: February 25, 2014

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1503.08303.

For every pair (G, V ) where G is a connected simple
linear algebraic group and V is a simple algebraic G-module with
a free algebra of invariants, the number of irreducible components
of the nullcone of unstable vectors in V is found. ...

Added: March 31, 2015

Roman Avdeev, Cupit-Foutou S., Advances in Mathematics 2018 Vol. 328 P. 1299-1352

Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme M_Γ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an `adjoint torus' T_ad and has a distinguished T_ad-fixed point X_0. ...

Added: February 25, 2018

Roman Avdeev, Transformation Groups 2021 Vol. 26 No. 2 P. 403-431

Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, ...

Added: September 9, 2021

V. L. Popov, Transformation Groups 2011 Vol. 16 No. 3 P. 827-856

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a
closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove
that in arbitrary G such a cross-section exists if and only if the ...

Added: March 16, 2013

Arzhantsev I., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

V. L. Popov, Doklady Mathematics 2017 Vol. 96 No. 1 P. 312-314

For connected simple algebraic groups defined over an algebraically closed field of characteristic
zero, the classifications of irreducible algebraic representations of modalities 0, 1, and 2 are obtained. ...

Added: June 30, 2017

Roman Avdeev, Petukhov A., Transformation Groups 2021 Vol. 26 No. 3 P. 719-774

Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to ...

Added: September 2, 2020

Skripchenko A., Dynnikov I., American Mathematical Society Translations: Series 2 2014 No. 234 P. 173-199

It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is self-similar, then a typical leaf has exactly one topological end. We also construct ...

Added: March 3, 2014

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1504.03867.

Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem is the affirmative. To this end we ...

Added: April 16, 2015

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.

Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...

Added: May 28, 2021

Roman Avdeev, Indagationes Mathematicae 2012 Vol. 23 No. 1-2 P. 10-18

In 1994, based on Roberts’ counterexample to Hilbert’s fourteenth problem, A’Campo-Neuen constructed an example of a linear action of a 12-dimensional commutative unipotent group H_0 on a 19-dimensional vector space V such that the algebra of invariants k[V]^{H_0} is not finitely generated. We consider a certain extension H of H_0 by a one-dimensional torus and ...

Added: February 25, 2014