Sofic profile and computability of Cremona groups

Cornulier, Yves de

doi:10.1307/mmj/1387226167

Cited by 15 publications

(22 citation statements)

References 46 publications

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“…In [9] are given examples of finitely generated (and even finitely presented) groups that do not admit embeddings into any Cremona group, which gives an answer to Cantat's question about the existence of such groups (see also [7]). These examples are based on Theorem 1.2 of [10], according to which the word problem is solvable in every finitely generated subgroup of any Cremona group. However, the answer to the above question -and even in a stronger form, with the addition of the condition of simplicity of the subgroup -can be obtained without using this theorem.…”

Section: Corollaries 1 and 2 Formulated Below Follow From Theorem 1 I...mentioning

confidence: 99%

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Embeddings of groups ${\rm Aut}(F_n)$ into automorphism groups of algebraic varieties

Popov¹

2021

Preprint

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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 2, such groups Aut(X) are nonamenable, and for n 3, they are nonlinear and contain the braid group B n . Some of these varieties X are affine, and among affine, some are rational and some are not, some are smooth and some are singular. The byproduct is that for n 3, each Cremona group of rank 3n contains Aut(F n ) and the braid group B n .

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Section: Corollaries 1 and 2 Formulated Below Follow From Theorem 1 I...mentioning

confidence: 99%

“…Thus, in view of (2), (3), (10), (11), we have σ X = id. Hence, σ is a nonidentity element of the kernel of homomorphism (6).…”

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confidence: 97%

Embeddings of groups ${\rm Aut}(F_n)$ into automorphism groups of algebraic varieties

Popov¹

2021

Preprint

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“…A weakness of Definition 2.3 is that it depends on the presentation of G. Following [Cor13], we can define another version of hyperlinear profile which is independent of the presentation, while still being in the spirit of Definition 2.3. Given a finite subset E of G containing the identity, let σ(E; δ, ǫ) be defined similarly to η(E; δ, ǫ), but with U(C d ) replaced by the group S d of d × d permutation matrices.…”

Section: A Presentation-independent Version Of Hyperlinear Profilementioning

confidence: 99%

“…Given a finite subset E of G containing the identity, let σ(E; δ, ǫ) be defined similarly to η(E; δ, ǫ), but with U(C d ) replaced by the group S d of d × d permutation matrices. Then σ(E) is a two-parameter version of the sofic profile defined in [Cor13]. Specifically, the sofic profile of E is (7.1) N → N ∪ {+∞} : n → σ E; 2 − 2 n , 2 n .…”

Section: A Presentation-independent Version Of Hyperlinear Profilementioning

confidence: 99%

Entanglement in Non-local Games and the Hyperlinear Profile of Groups

Slofstra

Vidick

2018

Ann. Henri Poincaré

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We relate the amount of entanglement required to play linearsystem non-local games near-optimally to the hyperlinear profile of finitelypresented groups. By calculating the hyperlinear profile of a certain group, we give an example of a finite non-local game for which the amount of entanglement required to play ǫ-optimally is at least Ω(1/ǫ k ), for some k > 0. Since this function approaches infinity as ǫ approaches zero, this provides a quantitative version of a theorem of the first author.

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“…cantat and j. xie group. One does not know whether groups of birational transformations share the same properties (see [14] and [21]). The following result implies Theorem C of the introduction.…”

Section: Central Extensions and Simple Groupsmentioning

confidence: 99%

Algebraic actions of discrete groups: the $p$-adic method

Cantat

Xie

2018

Acta Mathematica

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The first argument is based on p-adic analysis and may be viewed as an extension of two classical strategies from a linear to a non-linear context. The first strategy appeared in the proof of the theorem of Skolem, Mahler, and Lech, which says that the zeros of a linear recurrence sequence occur along a finite union of arithmetic progressions. This method plays now a central role in arithmetic dynamics (see [6] , [7], [55]). The second strategy has been developed by Bass, Milnor, and Serre when they obtained rigidity results for finite-dimensional linear representations of SL n (Z) as a corollary of the congruence subgroup property (see [2], [62]). Here, we combine these strategies for non-linear actions of finitely generated groups of birational transformations.Our second argument makes use of isoperimetric inequalities from geometric group theory and of the Lang-Weil estimates from diophantine geometry. We now list the main results that follow from the combination of those arguments. Actions of SL n (Z)Consider the group SL n (Z) of n×n matrices with integer entries and determinant 1. Let Γ be a finite-index subgroup of SL n (Z); it acts by linear projective transformations on the projective space P n−1 C , and the kernel of the homomorphism Γ!PGL n (C) contains at most two elements. The following result shows that Γ does not act faithfully on any smaller variety.

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Sofic profile and computability of Cremona groups

Cited by 15 publications

References 46 publications

Embeddings of groups ${\rm Aut}(F_n)$ into automorphism groups of algebraic varieties

Embeddings of groups ${\rm Aut}(F_n)$ into automorphism groups of algebraic varieties

Entanglement in Non-local Games and the Hyperlinear Profile of Groups

Algebraic actions of discrete groups: the $p$-adic method

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