Juan de Vicente - Uni KonstanzJuan de Vicente Universidad Aut onoma de Madrid Summer School in Tame...

Simply-connected locally Nash groups

Juan de Vicente

Universidad Autonoma de Madrid

Summer School in Tame Geometry,Konstanz, 2016

Juan de Vicente (UAM) Simply-connected locally Nash groups Tame Geometry, 2016 1 / 15

Outline

1 What is a locally Nash group?

2 Why do we study locally Nash groups?

3 A description of simply-connected two-dimensional abelian locally Nashgroups


Locally Nash Groups (LNGs)

Definition

Let G be a (real) Lie group with an analytic atlas A. Then G is a LNG if∃(U, ϕ) ∈ A, 1 ∈ U, for which (gU, ϕg )g∈G is an analytic atlas

ϕg : gU → ϕ(U) : gu 7→ ϕ(u)

s.t. its transition maps are Nash maps and ·,−1 (seen through the charts)are locally Nash maps. Notation:

(G , (U, ϕ)

)

∴ for (U, ϕ), ∃V ,Vg ⊂ U (g ∈ G ), 1 ∈ U, s.t.

(C1) ϕ(V )× ϕ(V ) −→ ϕ(U) : (x , y) 7→ ϕ(ϕ−1(x)ϕ−1(y))

(C2) ϕ(Vg ) −→ ϕ(U) : x 7→ ϕ(g−1(ϕ−1(x))−1g)

are Nash maps.


Locally Nash Groups (LNGs)

Definition

Let G be a (real) Lie group with an analytic atlas A. Then G is a LNG if∃(U, ϕ) ∈ A, 1 ∈ U, for which (gU, ϕg )g∈G is an analytic atlas

ϕg : gU → ϕ(U) : gu 7→ ϕ(u)

s.t. its transition maps are Nash maps and ·,−1 (seen through the charts)are locally Nash maps. Notation:

(G , (U, ϕ)

)∴ for (U, ϕ), ∃V ,Vg ⊂ U (g ∈ G ), 1 ∈ U, s.t.

(C1) ϕ(V )× ϕ(V ) −→ ϕ(U) : (x , y) 7→ ϕ(ϕ−1(x)ϕ−1(y))

(C2) ϕ(Vg ) −→ ϕ(U) : x 7→ ϕ(g−1(ϕ−1(x))−1g)

are Nash maps.


Definition

Let(G , (U, ϕ)

)and

(H, (V , ψ)

)be LNGs.

A homomorphism (isomorphism) of Lie groups f : G → H is a locally Nashhomomorphism (resp. isomorphism) if ∃W ⊂ U ∩ f −1(V ), 1 ∈W , s.t.

ϕ(W )→ ψ(V ) : x 7→ ψ f ϕ−1(x)

is a Nash map.


Example

1(R, (R, exp)

) exp'

(R>0, (R>0, id)

).

2(R, (R, sin)

).

3(R, (R, id)

) id'(R, ((−1, 1), id)

).

4 The direct product of LNGs is a LNG.

5 The universal covering of a LNG is a LNG.

6 The quotient of a LNG by a discrete subgroup is a LNG.

7 Every algebraic group has a natural structure of LNG.

Theorem (Hrushovski-Pillay [HP94])

The LNGs are, up to local Nash isomorphism, precisely the quotient ofuniversal coverings of algebraic groups by discrete subgroups.


IF(Rn, (U, ϕ)) is a LNG THEN

ϕ(V )× ϕ(V ) −→ ϕ(U) : (x , y) 7→ ϕ(ϕ−1(x) + ϕ−1(y))

ϕ(V )× ϕ(V ) −→ ϕ(U) : (ϕ(u), ϕ(v)) 7→ ϕ(u + v)

are Nash maps.

Algebraic Addition Theorem (AAT)

Let U be an open neighborhood of 0. An analytic map ϕ : U → Rn admitsan AAT if ϕ1(u + v), . . . , ϕn(u + v) are algebraic over

R(ϕ1(u), . . . , ϕn(u), ϕ1(v), . . . , ϕn(v)).

This definition generalizes to holomorphic and meromorphic maps.


Historical motivationNash groups and semialgebraic groups

A natural problem in Tame Geometry is to find a classification ofsemialgebraic groups over the real field, i.e. of groups definable over

R :=< R;<,+, ·, 0, 1 > .

In [Pil88], it is shown that these groups admit an additional analyticstructure and hence they can be viewed as Nash groups (and conversely,every Nash group is a semialgebraic group).

The one-dimensional classification of Nash groups was done by Maddenand Stanton in [MS92] by getting first a classification of simply-connectedone-dimensional LNG.


Notation

IF ϕ : Cn → Cn is a real meromorphic map admitting an AAT s.t.∃U ⊂ Rn, 0 ∈ U, ∃k ∈ Rn s.t.

ψ : U → Rn : x 7→ ϕ(x + k)

is a real analytic diffeomorphism THEN(Rn, (U, ψ)

)is a LNG.

Notation independent of U and k : (Rn, ϕ)

The Weierstrass ℘Λ-function ℘Λ : C→ C is a meromorphic functionadmitting an AAT which is not analytic at 0.An adecuate translation of ℘Ω is an analytic diffeomorphism at 0,∴ (R, ℘Λ) is a LNG.


Theorem (Madden-Stanton [MS92])

Classification of simply-connected one-dimensional LNGs:

(R, id),

(R, exp),

(R, sin),

(R, ℘Λ) for some lattice Λ < (C,+).

None of the above LNGs is locally Nash isomorphic to one of a differentclass. The fourth case subdivides in different cases.


Simply-connected two-dimensional abelian LNGs

Theorem (Baro-dV-Otero [BdVO15])

1 Extension: Every simply-connected two-dimensional abelian LNG islocally Nash isomorphic to one of the form (R2, ϕ) for some realmeromorphic map ϕ := (ϕ1, ϕ2) : C2 → C2 admitting an AAT.

2 Painleve families: Both ϕ1 and ϕ2 are algebraic over a field of aunique member of the six following families:


(ϕ1, ϕ2) : C2 → C2. Both ϕ1, ϕ2 are algebraic over one and only one of:

Painleve families

P1 :=C((u, v) α) : α ∈ GL2(C

)P2 :=

C((u, ev ) α) : α ∈ GL2(C

)P3 :=

C((eu, ev ) α) : α ∈ GL2(C

)

P4 :=C((℘Ω(u), v − aζΩ(u)) α

): a ∈ 0, 1; Ω < (C,+);α ∈GL2(C)

P5 :=

C((℘Ω(u), σΩ(u−a)

σΩ(u) ev) α)

: a ∈ C; Ω < (C,+);α ∈ GL2(C)

P6 : All the possible fields of abelian functions (corresponding to latticesof C2) of transcendence degree 2 over C.


(ϕ1, ϕ2) : C2 → C2. Both ϕ1, ϕ2 are algebraic over one and only one of:

Painleve families

P1 :=C((u, v) α) : α ∈ GL2(C

)P2 :=

C((u, ev ) α) : α ∈ GL2(C

)P3 :=

C((eu, ev ) α) : α ∈ GL2(C

)P4 :=

C((℘Ω(u), v − aζΩ(u)) α

): a ∈ 0, 1; Ω < (C,+);α ∈GL2(C)

P5 :=

C((℘Ω(u), σΩ(u−a)

σΩ(u) ev) α)

: a ∈ C; Ω < (C,+);α ∈ GL2(C)

P6 : All the possible fields of abelian functions (corresponding to latticesof C2) of transcendence degree 2 over C.


Proof: Extension

Proposition (Baro-dV-Otero [BdVO15])

IF (Rn, (U, ϕ)) is a LNG THEN ∃ψ := (ψ1, . . . , ψn) : Cn → Cn a realmeromorphic map admitting an AAT s.t.

1(Rn, (U, ϕ)) = (Rn, ψ),

2 ∃ψ0 : Cn → C a real meromorphic map algebraic over R(ψ1, . . . , ψn)s.t. for each f ∈ R(ψ0, ψ1, . . . , ψn)

f (u + v) ∈ R(ψ0(u), ψ1(u) . . . , ψn(u), ψ0(v), ψ1(v) . . . , ψn(v)

).

∴ the classification of locally Nash structures of R2 reduces to the study ofreal meromorphic maps ψ : C2 → C2 that admit an AAT.


Proof: Painleve families

Theorem (Painleve [Pai03])

IF ϕ1, ϕ2 : C2 → C are functionally independent meromorphic functionss.t. ϕ := (ϕ1, ϕ2) admits an AAT THEN ∃i ∈ 1, . . . , 6 s.t. ϕ1 and ϕ2

are algebraic over one of the fields of the family Pi .


Bibliography I

E. Baro, J. de Vicente, and M Otero.Two-dimensional simply connected abelian locally Nash groups.http://arxiv.org/abs/1506.00405, 2015.

Ehud Hrushovski and Anand Pillay.Groups definable in local fields and pseudo-finite fields.Israel J. Math., 85(1-3):203–262, 1994.

James J. Madden and Charles M. Stanton.One-dimensional Nash groups.Pacific J. Math., 154(2):331–344, 1992.

Paul Painleve.Sur les fonctions qui admettent un theoreme d’addition.Acta Math., 27(1):1–54, 1903.


Bibliography II

Anand Pillay.On groups and fields definable in o-minimal structures.Journal of Pure and Applied Algebra, 53(3):239–255, 1988.

Masahiro Shiota.Nash manifolds, volume 1269 of Lecture Notes in Mathematics.Springer-Verlag, Berlin, 1987.


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