Actions of Cremona groups on CAT(0) cube complexes

Lonjou, Anne; Urech, Christian

doi:10.48550/arxiv.2001.00783

Cited by 7 publications

(10 citation statements)

References 18 publications

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“…In [LU20], the second and third author constructed for every surface S the blow-up complex -a CAT(0) cube complex, on which Bir(S) acts by isometries -and used it to deduce various dynamical and group theoretical properties. However, one of the drawbacks of this construction is that these cube complexes are never locally compact.…”

Section: Cremona Groupsmentioning

confidence: 99%

“…Our main contribution to Neretin groups N d := AAut(T d ), where T d is the regular rooted tree of degree d ≥ 2, is the construction of locally compact CAT(0) cube complexes on which they act properly (as topological groups). Even if the existence of proper actions on (a priori not locally compact) CAT(0) cube complexes was already known (see [LB15, Section 3.3.3]), transfering the cubulation of Cremona groups from [Lon17,LU20] into the world of Nereting groups allows us to construct explicit cube complexes, whose geometric structures are tightly connected to the algebraic structures of Neretin groups.…”

Section: Introductionmentioning

confidence: 99%

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Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Genevois¹,

Lonjou²,

Urech³

2021

Preprint

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We show that plane Cremona groups over finite fields embed into Neretin groups, i.e., groups of almost automorphisms of rooted trees. The image of this embedding is dense if our base field has odd characteristic or is equal to F2. This is no longer true, if the finite base field has even characteristic and contains at least 4 elements; for this case we show that the permutations induced by birational transformations on rational points of regular projective surfaces are always even. In a second part, we construct explicit locally compact CAT(0) cube complexes, on which Neretin groups act properly. This allows us to recover in a unified way various results on Neretin groups such as that they are of type F∞. We also prove a new fixed-point theorem for CAT(0) cube complexes without infinite cubes and use it to deduce a regularisation theorem for plane Cremona groups over finite fields.2010 Mathematics subject classification.

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Section: Cremona Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Genevois¹,

Lonjou²,

Urech³

2021

Preprint

Self Cite

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“…This is interesting, considering the following facts. There are (infinite-dimensional) hyperbolic space and cubical complexes, on which Bir(P 2 k ) acts isometrically (see [21], Section 3.1.2 and [45]). The Cremona group Bir(P 2 k ) is sub-quotient universal: every countable group can be embedded in a quotient group of Bir(P 2 k ) (see [21], Theorem 4.7).…”

Section: Introductionmentioning

confidence: 99%

Length functions on groups and rigidity

Ye¹

2021

Preprint

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Let G be a group. A function l : G → [0, ∞) is called a length function if (1) l(g n ) = |n|l(g) for any g ∈ G and n ∈ Z;(2) l(hgh −1 ) = l(g) for any h, g ∈ G; and(3) l(ab) ≤ l(a) + l(b) for commuting elements a, b. Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on CAT(0) spaces and Gromov δ-hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, dynamical degrees of birational maps and so on. We study length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups over rings and Cremona groups. As applications, we prove that every group homomorphism from an arithmetic subgroup of a simple algebraic Q-group of Q-rank at least 2, or a finite-index subgroup of the elementary group E n (R) (n ≥ 3) over an associative ring, or the Cremona group Bir(P 2 C ) to any group G having a purely positive length function must have its image finite. Here G can be outer automorphism group Out(F n ) of free groups, mapping classes group MCG(Σ g ), CAT(0) groups or Gromov hyperbolic groups, or the group Diff(Σ, ω) of diffeomorphisms of a hyperbolic closed surface preserving an area form ω.

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“…In dimension 2, it is understood only in the case of birational maps (Diller-Favre [DF01]), of polynomial maps (Favre-Jonsson [FJ11]) and of toric maps (Lin [Lin12] and Favre-Wulcan [FW12]). Partial results on degree growths for arbitrary rational maps have been obtained by Urech [Ure18], Cantat-Xie [CX18], and recently by Lonjou-Urech [LU20].…”

Section: Introductionmentioning

confidence: 99%

Spectral interpretations of dynamical degrees and applications

Dang

Favre

2020

Preprint

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We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space of b-classes of higher codimension cycles, and endow this space with various Banach norms. Building on these constructions, we design a natural extension to higher dimension of the Picard-Manin space introduced by Cantat and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the Hodge index theorem, and a surprising compactness result in this Banach space. We use these two theorems to infer a precise control of the sequence of degrees of iterates of a map under the assumption λ 2 1 > λ 2 on the dynamical degrees. As a consequence, we obtain that the dynamical degrees of an automorphism of the affine 3-space are all algebraic numbers. Contents5 The spectral gap on N 1 Σ (X ) when λ 1 (f ) 2 > λ 2 (f ) 43 5.

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Actions of Cremona groups on CAT(0) cube complexes

Cited by 7 publications

References 18 publications

Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Length functions on groups and rigidity

Spectral interpretations of dynamical degrees and applications

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