Definable Equivariant Retractions in Non-Archimedean Geometry
aa r X i v : . [ m a t h . L O ] J a n DEFINABLE EQUIVARIANT RETRACTIONS INNON-ARCHIMEDEAN GEOMETRY
MARTIN HILS, EHUD HRUSHOVSKI, AND PIERRE SIMON
Abstract.
For G an algebraic group definable over a model of ACVF, ormore generally a definable subgroup of an algebraic group, we study the stablecompletion b G of G , as introduced by Loeser and the second author. For G connected and stably dominated, assuming G commutative or that the val-ued field is of equicharacteristic 0, we construct a pro-definable G -equivariantstrong deformation retraction of b G onto the generic type of G .For G = S a semiabelian variety, we construct a pro-definable S -equivariantstrong deformation retraction of b S onto a definable group which is internal tothe value group. We show that, in case S is defined over a complete valuedfield K with value group a subgroup of R , this map descends to an S ( K )-equivariant strong deformation retraction of the Berkovich analytification S an of S onto a piecewise linear group, namely onto the skeleton of S an . Thisyields a construction of such a retraction without resorting to an analytic(non-algebraic) uniformization of S .Furthermore, we prove a general result on abelian groups definable in anNIP theory: any such group G is a directed union of ∞ -definable subgroupswhich all stabilize a generically stable Keisler measure on G . Contents
1. Introduction 22. Tying up some loose ends 73. Existence of a definable equivariant retraction 154. An explicit definable equivariant retraction in equicharacteristic 0 185. Application to the topology of S an Date : January 8, 2021.2020
Mathematics Subject Classification.
Primary: 03C45; Secondary: 03C98, 12J25, 14G22.
Key words and phrases.
Model Theory, Stably Dominated Group, Stable Completion, Semia-belian Variety, Non-Archimedean Geometry.MH was partially supported by the German Research Foundation (DFG) via CRC 878, HI2004/1-1 (part of the French-German ANR-DFG project GeoMod) and under Germany’s Excel-lence Strategy EXC 2044-390685587, ‘Mathematics M¨unster: Dynamics-Geometry-Structure’.PS was partially supported by NSF grants no. 1665491 and 1848562.All three authors were supported by the Hebrew University of Jerusalem and by the Euro-pean Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant agreement no. 291111/MODAG. Introduction
In [9], Loeser and the second author have developed a novel model-theoreticapproach to non-Archimedean (algebraic) geometry. For an algebraic variety V defined over some valued field K , they construct the so-called stable completion of V , a pro-definable space, more precisely a functor b V which is pro-definable over K and which associates to any valued field extension L of K a topological space b V ( L )naturally containing V ( L ), endowed with the valuation topology, as a subspace. Incase L is non-trivially valued and algebraically closed, V ( L ) is dense in b V ( L ).The construction of b V is similar to that of the Berkovich analytification V an of V ([2], see also [4]), which is only defined when K is complete with value group a sub-group of R . As is the case for V an , the stable completion b V has good topologicalproperties, e.g., analogues of local (pathwise) connectedness and local compact-ness hold for b V in the definable category. Assuming that V is quasi-projective,strong topological tameness properties are established for b V in [9]. Most notably,it is shown that b V admits a pro-definable strong deformation retraction onto a Γ-internal space, i.e., onto a piecewise linear space in the definable category. Here,Γ denotes the value group. This parallels topological tameness results in the caseof V an , established by Berkovich. Actually, assuming K is complete with Γ K ≤ R ,it is shown in [9, Chapter 14] that, for some suitable K max ⊇ K , V an may beobtained as a topological quotient of b V ( K max ) and the pro-definable strong defor-mation retraction descends to a strong deformation retraction of V an onto a finitesimplicial complex, thus reproving and generalizing results by Berkovich, as thisholds for every quasi-projective variety V , without any smoothness assumption on V . Moreover, in this context, local contractibility of V an is established in [9], aswell as the finiteness of the number of homotopy types in {V an b | b ∈ S ( K ) } , where V → S is a quasi-projective family of algebraic varieties defined over K . For a sur-vey of the results in [9] and in particular the consequences in the realm of Berkovichspaces, we refer to [5].The main tools used in [9] concern the geometric model theory of the (first order)theory ACVF of algebraically closed non-trivially valued fields. The study of ACVFgoes back to Abraham Robinson who established its model-completeness. His argu-ments actually yield quantifier elimination for ACVF in various natural languages.Haskell, Macpherson and the second author initiated the study of ACVF from apoint of view of geometric model theory. In [6], they show that the imaginary sortsin ACVF are classified by certain natural group quotients, the so-called geometricsorts , and they identify the stable part of ACVF as the collection of definable setswhich are internal in the residue field k . In [7] the authors systematically developstable domination as a means to lift phenomena known in stable theories to theunstable context. Stably dominated types are definable types whose generic ex-tension is entirely controlled by its stable part. In ACVF, these correspond to thedefinable types orthogonal to Γ.Let us briefly describe the construction of the Berkovich analytification of analgebraic variety. Suppose that K is a field which is complete with respect to anon-Archimedean norm | · | : K → R ≥ and that V is an algebraic variety definedover K . The construction of the anaytification V an is done by gluing affine pieces,so let us assume that V is affine. Then V an is the set of all multiplicative semi-norms | · | v : K [ V ] → R ≥ extending | · | on K , and the topology on V an is defined as EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 3 the weakest topology such that for any f ∈ K [ V ], the map | · | v f | v is continuous.In this way, the natural embedding of V ( K ) into V an becomes a homeomorphism.Identifying R ≥ with R ∪{∞} via − log, the field K becomes a (complete) valuedfield with Γ K ≤ R , and, by quantifier elimination in ACVF, as a set, V an equals { p = tp( a/K ∪ R ) ∈ S V ( K ∪ R ) | Γ K ( a ) ≤ R } , where S V ( K ∪ R ) denotes the set of complete types concentrating on V , in thetheory ACVF, over the (2-sorted) parameter set K ∪ R .The key insight of [9] is that one may obtain a model-theoretic analogue of V an by considering those (global definable) types concentrating on V whose realizationsdo not increase the value group, relative to the base model over which one works.Consequently, the stable completion b V is defined as the set of stably dominatedtypes concentrating on V , endowed with a topology which is similarly defined as inthe case of V an . It is an important feature that, when one identifies a definable typewith its canonical base, b V naturally gets the structure of a (strict) pro- K -definablespace in ACVF. Thus model-theoretic methods may be applied to b V , in particularpowerful tools both from stability theory and from o -minimality, two well developedand completely different strands of geometric model theory. Any type tp( a/K ) withtr( K ( a ) /K ) = tr( k K ( a ) /k K ) is stably dominated, and the collection of these types,which corresponds to the sub-collection of all strongly stably dominated types,forms an ind-definable subspace V of b V .Now let us move to the equivariant situation. If K is a complete field with re-spect to an non-Archimedean norm and G is an algebraic group defined over K , then G ( K ) naturally acts on G an , and one may ask whether there is a G ( K )-equivariantretraction of G an onto a skeleton which is then naturally a piecewise linear group.For E a Tate elliptic curve defined over K , i.e., an elliptic curve with bad (split)multiplicative reduction, E an does indeed admit an E ( K )-equivariant strong defor-mation retraction onto the circle group R / Z , as one may for example show usingthe Tate uniformization; in case E is an elliptic curve with good reduction, E an equivariantly retracts to the trivial group. Actually, whenever S is an abelian oreven semi-abelian variety defined over K , passing to a finite separable extension of K if necessary, S an admits an analytic uniformization by an analytic group whichis equivariantly contractible, from which one obtains an S ( K )-equivariant strongdeformation retraction of S an onto its skeleton (see [2, Section 6.5], where this isexplained for abelian varieties).In the present paper, we prove analogous results for the stable completion b S of asemiabelian variety S defined over some valued field K , confirming that the model-theoretic approach to non-Archimedean geometry developed in [9] is very adequatefor the study of semiabelian varieties as well, as the topological tameness propertiesone expects do also hold in this context. We would like to highlight that, contrarilyto the classical approach via analytic uniformizations (see [3]), our constructionmay be performed over an arbitrary valued field (with value group not necessarilyArchimedean) and does not require the passage to a finite separable extension inthe process. Moreover, we reprove the results for S an mentioned in the previousparagraph in the same way topological tameness results for V an could be deducedin [9] from the corresponding results for the stable completion b V in the definablecategory. MARTIN HILS, EHUD HRUSHOVSKI, AND PIERRE SIMON
Before we may state the main contributions of our paper, we need to mention aresult from [12], which is a crucial ingredient in our construction.In [7], an important and very useful structural way to decompose types in ACVFis obtained: if M is a maximally complete model of ACVF and a is a (possibly imag-inary) tuple from the monster model U , then tp( a/M ∪ Γ( M a )) is stably dominated.So types may be understood, to some extent, by types in the value group and typesin the residue field. Moreover, ACVF has the invariant extension property, i.e.,any type over an algebraically closed set admits a global automorphism invariantextension. It follows that ACVF is a metastable theory.Metastability is an axiomatic framework capturing the phenomena we describedin the previous paragraph. It has been introduced in [7] and further developedin work of Rideau-Kikuchi and the second author [12], who undertake a thoroughstudy of groups definable in metastable theories. In their work, stably dominatedgroups, i.e., ( ∞ -)definable groups admitting a stably dominated generic type, playa key role, an important reason being that many features of stable groups lift tothem. As one of the main results, they establish a group version of the metastabilityproperty in the commutative case: for any definable abelian group G in ACVF,there is a definable homomorphism λ : G ։ Λ such that Λ is Γ-internal and N := ker( λ ) is a connected group which is an increasing union (indexed essentiallyby the value group) of definable stably dominated subgroups.We show that whenever G = S is a semiabelian variety, N is in fact itself stablydominated. More precisely, if S is defined over a valued field F ⊆ K | = ACVF,there is an F -definable decomposition(1.1) 0 → N → S → Λ → N stably dominated, definable and connected. Inparticular, it follows that p N ∈ b S , where p N denotes the generic type of N .For abelian varieties, stable domination of N is shown in [12]. It is easy tosee that this also holds for an algebraic torus, e.g., if G = G nm , then N = ( O ∗ ) n is stably dominated, with G/N ∼ = Γ n . We show that stable domination of thecorresponding N is preserved in short exact sequences of algebraic groups. As asemiabelian variety is an extension of an abelian variety by an algebraic torus, wemay thus lift the result to the semiabelian case.For G any definable group, we denote by p e ∈ b G the type of the identity elementin G . We obtain the following result. Theorem A (Theorem 3.10) . Let S be a semi-abelian variety defined over F ⊆ K | = ACVF , and let → N → S → Λ → be the decomposition from (1.1) .Then there is an F -definable special deformation retraction ρ : [0 , ∞ ] × b S → b S with final image Σ ⊆ b S such that ρ is equivariant under the action of S bymultiplication and for each t < ∞ , q t = ρ ( t, p e ) is the generic type of a connectedstrongly stably dominated definable subgroup N t of N , with N t Zariski dense in S .Moreover, the morphism π : S → Λ induces definable bijection between Σ and Λ . Modulo some continuity issues concerning the map given by the tensor product ⊗ (dealt with in Subsection 2.6), Theorem A follows in a rather straight forward This may be seen as a very powerful incarnation of the Ax-Kochen-Ershov philosophy whichpostulates that model-theoretic questions about henselian valued fields may be reduced to ques-tions about the residue field and the value group. See Definition 3.1 for the notion of a special deformation retraction.
EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 5 way from the existence of a definable continuous path in b N between p e and p N (thegeneric type of N ), along generic types of (Zariski-dense strongly stably dominated)definable subgroups N t of N . Let us illustrate this first with two easy examples.Denote by η c,γ the generic type of the closed ball B ≥ γ ( c ), for γ ∈ Γ ∪ {∞} . • The standard deformation retraction ρ : [0 , ∞ ] × b O → b O , sending ( γ, η c,δ )to η c, min( δ,γ ) is definable and ( O , +)-equivariant with final image { η , } ,and as an equivariant continuous map it is entirely determined by the path q : [0 , ∞ ] → b O , q ( γ ) := ρ ( γ, p e ) = η ,γ , where p e = η , ∞ . Thus, N γ = γ O in this example. • The map ρ ′ : [0 , ∞ ] × G m → d G m , ( γ, c ) η c,v ( c )+ γ extends uniquely to a G m -equivariant deformation retraction ρ : [0 , ∞ ] × d G m → d G m , via ρ ( γ, η c,v ( c )+ δ ) = η c,v ( c )+min( γ,δ ) (for c = 0 and δ ≥ . Its final image is { η c,v ( c ) | c = 0 } = { η ,γ | γ ∈ Γ } ∼ = Γ, and one has p e = η , ∞ , N = N = O ∗ and N γ = 1 + γ O for all γ ∈ (0 , ∞ ). Setting q γ = ρ ( γ, p e ) = η ,γ , one may check that ρ ( γ, p ) = b m ( q γ ⊗ p ) , the convolution of q γ and p . Here, m denotes the multiplication in G m .In our paper, we give two proofs of the existence of a group path as above, onein arbitrary characteristic which is valid in all stably dominated abelian definablesubgroups of an algebraic group, one in equicharacteristic 0, valid for all stablydominated subgroups of an algebraic group: Theorem B (Theorem 3.6 & Theorem 4.17) . Let G be an algebraic group definedover a valued field F , and let N be an F -definable stably dominated connectedsubgroup of G . Assume that N is commutative or that F is of equicharacteristic 0.Then there is an F -definable path q : [0 , ∞ ] → N such that(i) q ∞ = p e and q = p N ;(ii) for any t < ∞ , q t is the generic type of a connected stably dominateddefinable subgroup N t of N which is Zariski dense in N . The construction in arbitrary characteristic relies on the existence of a defin-able path between p e and p N , which is then turned into a group path, using anaveraging process (described in [12] and requiring commutativity of the group, fortechnical reasons) reminiscent of Zilber’s Indecomposability Theorem. The secondconstruction, only valid in equicharacteristic 0, is more explicit and does not re-quire the group N to be commutative. It utilizes an intrinsic scale provided bythe value group, and the subgroup N t in the group path is found as the kernelof the homomorphism to the maximal O /t O -internal quotient of N . The proofmay be understood as a linearization procedure at the level of generic types, whichis available even if, in ACVF, there exists no exponential map in the definablecategory.As we have already mentioned, we deduce Theorem A from Theorem B. More-over, Theorem B yields the following strong contractibility result as a corollary. Theorem C (Theorem 3.7 & Corollary 4.18) . Let N be a group which satisfies theassumptions of Theorem B. Then b N admits an F -definable N -equivariant specialdeformation retraction with final image { p N } . MARTIN HILS, EHUD HRUSHOVSKI, AND PIERRE SIMON
It is worth mentioning the following connection to group schemes over O . Assumethe base field F is algebraically closed. If H is a connected Zariski-dense stablydominated ( ∞ -definable) subgroup of the affine algebraic group G , there is a groupscheme H over O such that H is definably isomorphic to the group of O -valuedpoints of H . (See [12, Theorem 6.11].) Under additional assumptions on H , a chainof stably dominated subgroups becomes visible geometrically, namely via the mapval( c ) ker( H ( O ) → H ( O /c O )).Assume now that F is a valued field with value group Γ F ≤ R and that V isan algebraic variety defined over F . In [9, Chapter 14], strong topological linksare established between the stable completion b V of V (the definable world) and theBerkovich analytification V an (the analytic world). If F max denotes an algebraicallyclosed valued field extension of F with value group Γ F max = R and residue field k F max = ( k F ) alg , then there is a natural continuous surjective and closed map π V : b V ( F max ) → V an . Moreover, considering the parameter set F := ( F, R ), anypro- F -definable (special) deformation retraction H : [0 , ∞ ] × b V → b V descends to a (special) deformation retraction e H : R + ∞ × V an → V an , where R + ∞ = [0 , ∞ ]( F ) = R ≥ ∪{∞} . If the final image Z of H is Γ-internal, then thefinal image of e H equals Z = π V ( Z ( F max )), a set which is naturally homeomorphicto a piecewise linear subset of R n ∞ (where R ∞ = R ∪ {∞} and n ∈ N ), carrying a Q -tropical structure .We call such deformations e H definably induced . In case V = G is an algebraicgroup and H is G -equivariant, the map e H will automatically be G ( F )-equivariant.Theorem A thus yields the following topological tameness result about analytifica-tions of semi-abelian varieties. Theorem D (Theorem 5.4) . Let S be a semi-abelian variety defined over a valued F with Γ F ≤ R . Consider the F -definable decomposition → N → S → Λ → from (1.1) . Then there is a definably induced S ( F ) -equivariant special deformationretraction ˜ ρ : R + ∞ × S an → S an with final image a skeleton Σ . The map π : S → Λ induces a bijection between Σ and Λ( F ) , where Λ( F ) is a subset of R n carrying the structure of a piecewise linearabelian group. Let us mention that in case A is an abelian variety over a valued field F withΓ F ∼ = Z and if N ≤ A is as in (1.1), then N ( F ) corresponds to the O F -pointsof the identity component of the N´eron model A of A . Observe, though, thatthe (maximal) stably dominated subgroup N of A always exists, even when A isdefinable over a valued field with non-Archimedean value group.In the present article, even if we recall the relevant key concepts and results,we assume the reader is familiar with the model theory of ACVF as developed in[6, 7] and more specifically with the model-theoretic approach to non-Archimedean This means that all the coefficients in the inequalities describing Z may be taken from Q . We thank Antoine Ducros and Peter Schneider for pointing this out.
EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 7 geometry from [9]. Moreover, we will use results from [12] on stably dominatedgroups at many places, and so some acquaintance with these is certainly helpful.Here is an overview of the article. In Section 2, we gather a number of facts whichwe will use in our paper, in particular on stable domination, the stable completionof an algebraic variety and on stably dominated groups in ACVF. The materialwe present is mostly from [7, 9, 12], but some facts are new, in particular one onthe maximal internal quotient of a generically stable group (Proposition 2.9) and acontinuity result concerning ⊗ in the stable completion (Proposition 2.23), whichare both key ingredients in our construction.In Section 3, we give our first construction of a definable group path, valid in allcharacteristics, and in Section 4, we present the explicit construction in equichar-acteristic 0. The results about equivariant retractions are then transferred to thesetting of Berkovich analytifications in Section 5, where we prove in particularTheorem D. Finally, in Section 6, we prove a general result on commutative groupsdefinable in an arbitrary NIP theory (Proposition 6.3), which is a rather weakanalogue of our main result, namely that any such group G may be written asa directed union of ∞ -definable subgroups which all stabilize a generically stableKeisler measure on G . 2. Tying up some loose ends
We work in a complete theory T , and U denotes a monster model of T . Weassume throughout that T eliminates quantifiers and imaginaries. Our notationand terminology is mostly standard. If D is a C -definable set, for C a subset of U , we denote by D ( C ) the set D ( U ) ∩ dcl( C ). We will use various generalizationsof definable sets in our paper, in particular (strict) pro-definable and ∞ -definablesets, relatively definable subsets of such sets, and occasionally ind-definable andiso-definable sets. See [9, Section 2.2.] or [12] for definitions and properties of thesenotions. For basic facts about the model theory of ACVF, we refer to [6, 7].2.1. Stably embedded sets.
Let C be a small subset of U . Recall that a (small)family ( X i ) i ∈ I of C -definable sets in U is stably embedded if for any finite sequence( i , . . . , i m ) of elements of I , any U -definable subset of Q mj =1 X i j is definable withparameters from C ∪ S i ∈ I X i . If X and D are definable sets, X is said to be D -internal if ( X = ∅ or) there are m ∈ N and a surjective U -definable function g : D m → X . Generically stable types are definable types that behave in some ways like typesin a stable theory. For a definition of generic stability in an arbitrary theory T ,see, e.g., [1]. We will only be concerned with generically stable types in NIP theo-ries, where there are various useful characterizations of this notion (see [15, Theo-rem 2.29]). Lemma 2.1.
Let p be a C -definable generically stable type, and D a C -definableset. Let f − be a definable family of functions to D . Then the set of p -germs ofinstances of f − is D -internal.Proof. Let X be the set of p -germs of instances of f − , so X is a definable set sincethe type p is definable. Let a , . . . , a n be a sufficiently long Morley sequence of p over C . For a parameter c , let g ( c ) = ( f c ( a ) , . . . , f c ( a n )) ∈ D n . Taking n largeenough, if g ( c ) = g ( c ′ ) the germs of f c and f c ′ are equal. Hence the map π from theimage of g to X sending g ( c ) to the p -germ of f c shows that X is D -internal. (cid:3) MARTIN HILS, EHUD HRUSHOVSKI, AND PIERRE SIMON
Lemma 2.2. ( T is any theory.) Let X be an ∅ -definable set. Assume that X isstably embedded, then so is Int( X ) : the union of the ∅ -definable X -internal sets.Proof. First, let Y be ∅ -definable and f a : X → Y a definable bijection. Considerthe ∅ -definable set A of codes of bijections f a ′ : X → Y . For f ∈ A , define g = f − ◦ f a . Then g : X → X is a bijection. As X is stably embedded, g = h b forsome definable function h b , with b in some X n . The function mapping b to f suchthat h b = f − ◦ f a is a -definable and defines a surjection from some definable subsetof X n to A . Hence A is X -internal. Now any definable subset of Y is definableusing a parameter from A and parameters from X (since X is stably embedded),therefore any definable subset of Y is definable with parameters from Int( X ).Now if Y is an ∅ -definable X -internal set, there is a definable surjection f a : X m → Y . Let A be the set of codes of surjections f a ′ : X m → Y . Then A is ∅ -definable and any definable subset of Y is definable with parameters in X and A .It remains to show that A is X -internal. The argument is the same as above: let f, f ′ ∈ A , then we can define the correspondence f ′− ◦ f := { ( x, x ′ ) ∈ X m × X m : f ( x ) = f ′ ( x ′ ) } . By stable embeddedness of X , any such correspondence can bewritten as h b for some b ∈ X k . Now fixing an element f ∈ A , we see that anyelement of A is definable from such a b ∈ X k along with f , hence A is X -internalas required. (cid:3) Stably dominated and strongly stably dominated types.
We brieflyrecall some facts around (strongly) stably dominated types. Given a set of param-eters C , we denote by St C the collection of all C -definable stable stably embeddedsets. For a tuple a , we denote by St C ( a ) the set dcl( Ca ) ∩ St C ( U ).In the following definition, non-forking | ⌣ is meant with respect to the stablemulti-sorted structure St C ( U ). Definition 2.3 ([7, 12]) . A type p = tp( a/C ) is called stably dominated if forany tuple b with St C ( a ) | ⌣ St C ( C ) St C ( b ), one has tp( a/ St C ( a )) ⊢ tp( a/ St C ( a ) b )(or equivalently, tp( b/ St C ( a )) ⊢ tp( b/Ca )). It is called strongly stably dominated if there exists a formula φ ( x ) ∈ tp( a/ St C ( a )) such that for any tuple b withSt C ( a ) | ⌣ St C ( C ) St C ( b ), one has φ ( x ) ⊢ tp( a/ St C ( a ) b ).Clearly, every strongly stably dominated type is stably dominated.By [9, Proposition 2.6.12], if p is a definable type based on some C = acl( C ) and p | A is strongly stably dominated for some A = acl( A ), then p | C is strongly stablydominated. Fact 2.4 ([7]) . Let p be a global definable type in some theory T .(1) If p is stably dominated, then it is generically stable.(2) In T = ACVF , p is stably dominated if and only if it is generically stableif and only if it is orthogonal to Γ . The following is a special case of [9, Proposition 8.1.2].
Fact 2.5.
We work in
ACVF . Let q be an A -definable type on a variety V . Then q is strongly stably dominated if and only if dim( q ) = dim( h ∗ q ) for some A -definablemap h into a stable sort; where dim( h ∗ q ) refers to Morley dimension. Recall that a global definable type p is called orthogonal to Γ if Γ( U a ) = Γ( U ) for a | = p . EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 9
The stable completion of an algebraic variety.
We work in ACVF, andwe will now briefly recall the notion of the stable completion of an algebraic varietyfrom [9]. Given an algebraic variety V defined over a valued field F , its stablecompletion b V is a functor which to every set B over which V is defined associatesthe set b V ( B ) of all global B -definable stably dominated types concentrating on V .It is naturally given by a strict pro- F -definable set, i.e., by a projective limit of F -definable sets with surjective transition functions.One endows b V with the following ind-definable topology: for any F ⊆ K | =ACVF, a basis of open subsets for b V ( K ) is given by sets of the form { p ∈ b U | g ∗ ( p ) ∈ Ω } , for U an open affine subvariety of V defined over K , Ω ⊆ Γ ∞ an opensubset and g = val ◦ G for some regular function G on U . Here, g ∗ ( p ) ∈ Γ ∞ ( K ) isdefined as g ( a ), where a | = p | K . As p is orthogonal to Γ, this is well-defined.If X ⊆ V is a definable subset, b X := { p ( x ) ∈ b V | p ( x ) | = x ∈ X } is a relativelydefinable subset of b V which is endowed with the subspace topology. One denotesby X ( B ) the set of all strongly stably dominated types in b X ( B ). Then X isnaturally an ind-definable set, and we endow it with the subspace topology.We refer to [9] for the definition and study of definable analogues of varioustopological properties, in particular definable compactness and definable (path)connectedness.2.4. Pro-definable groups and generic types.
We now recall the concept ofgenericity in the context of pro-definable groups, following [12, Section 3]. Let G be a pro- C -definable group, where C is some small parameter set. A global C -definable type p concentrating on G is called right generic in G over C if for all g ∈ G ( U ), the type g · p is C -definable. Left genericity is defined in a similar way,as is right genericity in a pro-definable principal homogeneous space under G . Aright or left generic type p of G is called symmetric if for any other global definabletype q concentrating on G one has p ⊗ q = q ⊗ p . Note that in an NIP theory, thisis equivalent to generic stability of p .If G admits a smallest pro- U -definable subgroup H of bounded index, H iscalled the strong connected component of G and denoted by G . Similarly, if theintersection H of all pro- U -definable subgroups of finite index in G has boundedindex in G , then H is called the connected component of G and denoted by G .The group G is called connected if G = G , and strongly connected if G = G .The following fact combines [12, 3.4, 3.9, 3.11 & 3.12]. Fact 2.6. ( T is any theory.) Let G be a pro-definable group.Assume that p is a right (resp. left) generic type of G . Then the following holds:(1) G is pro-definably isomorphic to a pro-limit of definable groups. In partic-ular, if G is ∞ -definable, it is an intersection of definable groups.(2) G exists and one has G = G = Stab( p ) , where Stab( p ) denotes the left(resp. right) stabilizer of p .Assume in addition that p is symmetric. Then the following holds:(3) Right and left generic types coincide in G , they are all symmetric, and G acts transitively on the set of generic types, which is bounded. We refer to [9] and also to [5] for details. See [12] for the definition of pro-definable and ∞ -definable groups. In particular, p is left and right generic, and G is connected if and onlyif p is the unique (right) generic type of G . In our work, mostly symmetric (right) generic types will play a role. We will callthem generic types in what follows.
Definition 2.7.
A pro-definable group is called generically stable ( stably domi-nated or strongly stably dominated , respectively) if it admits a generic type whichis generically stable (stably dominated or strongly stably dominated, respectively).Note that all parts of Fact 2.6 apply to all generically stable and thus in particularto all stably dominated groups. Lemma 2.8. ( T is any theory.) Let G and H ≤ G be pro-definable groups. Assumethat G is stably dominated and that H admits a right generic type. Let a be genericin G . Then a is right generic in the coset a · H over π ( a ) , where π : G → G/H isthe canonical projection.Proof.
By Fact 2.6, we may assume that a ∈ G . Let b be right generic in H over a , such that b ∈ H , and so b ∈ G as well. By symmetry of the generic type of G , a is generic in G over b , as is ab − . By symmetry again, b is right generic in H over ab − . Being a translate of b over ab − , a = ( ab − ) · b is right generic in a · H over ab − . (Note that a · H is definable over ab − .) (cid:3) The following proposition is a key ingredient for our second construction of de-finable equivariant retractions in Section 4.
Proposition 2.9.
Let G be a pro- C -definable generically stable connected group,and let D be a stably embedded C -definable set. There exists a pro- C -definablegroup g D internal to D and a pro- C -definable homomorphism g : G → g D whichis maximal in the sense that for any other pro- C -definable group homomorphism g ′ : G → g ′ D with g ′ D internal to D , g ′ factors through g .The generic of g D is interdefinable over C with dcl( Ca ) ∩ Int C ( D ) , where a is ageneric of G over C and Int C ( D ) is the union of C -definable, D -internal sets.Proof. We follow the proof of [12, Proposition 4.6] very closely. By Lemma 2.2,Int C ( D ) is stably embedded. Let p be the (by Fact 2.6 unique) generic type of G . For any a , let θ ( a ) enumerate dcl( Ca ) ∩ Int C ( D ). Fixing a ∈ G generic over C , consider the map f a : b θ ( ab ) which is a pro- C -definable map on G . As thegeneric p of G is generically stable, the p -germ ˜ a of f a is strong by [1, Theorem 2.2],which means that there is a function f ′ ˜ a such that c := f ′ ˜ a ( b ) = f a ( b ) for b | = p | Ca .On the other hand, the germ ˜ a lives in dcl( Ca ) ∩ Int C ( D ) by Lemma 2.1 hence wecan write f ′ θ ( a ) . Then by stable embeddedness, tp( c, θ ( a ) /Cθ ( b )) ⊢ tp( c, θ ( a ) /Cb ),so c ∈ dcl( θ ( a ) , θ ( b )). We can now write θ ( ab ) = c = F ( θ ( a ) , θ ( b )) for a | = p and b | = p | Ca .By the group chunk theorem ([12, Proposition 3.15]), there is a pro- C -definablegroup g D for which F coincides generically with the multiplication map and θ extends to a pro- C -definable homomorphism g : G → g D .Universality follows from the fact that we have taken all of dcl( Ca ) ∩ Int C ( D ) inthe image of θ . (cid:3) Remark 2.10.
In the situation of Proposition 2.9, if C ⊆ C ′ , denoting the groupscomputed using C and C ′ by g CD and g C ′ D , respectively, there is by construction acanonical surjective pro- C ′ -definable homomorphism g C ′ D → g CD . EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 11
Stably dominated groups in
ACVF . The following result is a consequenceof [12, Proposition 4.6 & Corollary 4.11].
Fact 2.11.
Let N be a definable stably dominated group in ACVF . Then N = N has finite index in N . Lemma 2.12.
Let N be a definable stably dominated group in ACVF . Then thegeneric types of N are strongly stably dominated.Proof. By Fact 2.11, we may assume that N is connected. Let C be a modelover which N is defined, and we work over C . By [12, Proposition 4.6], thereis a definable homomorphism h : N → g , where g is a definable group in St C such that the generics of N are stably dominated via h , i.e., if p is a generictype of N and a | = p | C , then for any tuple b with h ( a ) | ⌣ St C ( C ) St C ( b ), one hastp( b/Ch ( a )) ⊢ tp( b/Ca ).We show that tp( a/Ch ( a )) is isolated by the formula h ( x ) = h ( a ), from whichthe result follows. By [12, Lemma 4.9], we have that tp( a/C ) is generic in G if andonly if tp( h ( a ) /C ) is generic in g . Hence if h ( a ′ ) = h ( a ), then a ′ is generic in N over C , hence tp( a ′ /C ) = tp( a/C ), hence tp( a ′ h ( a ′ ) /C ) = tp( ah ( a ) /C ) and finallytp( a ′ /Ch ( a )) = tp( a/Ch ( a )) as required. (cid:3) Here is a partial converse to the preceding lemma.
Lemma 2.13.
Let N be an ∞ -definable strongly stably dominated connected sub-group of an algebraic group G in ACVF . Then N is definable.Proof. We may replace G by the Zariski closure of N in G and thus assume withoutloss of generality that N is Zariski dense in G . As the generic type of N is stronglystably dominated, definability of N follows from [12, Proposition 4.6 in combinationwith Corollary 4.16]. (cid:3) Lemma 2.14.
Let N be a definable stably dominated group in ACVF , . Assumethat N is abelian and connected. Then N is divisible.Proof. Let p be a prime number and assume that pN = N . Then pN has infiniteindex in N and the group N/pN is an infinite stably dominated abelian p -group.We know, e.g., by [12, Proposition 4.6], that this group maps to an infinite stablegroup g , where g is then also an abelian p -group. As g is internal to k , it isdefinably isomorphic to a group definable in ACF , whence definably isomorphicto an algebraic group. It is well known that there are no infinite commutativealgebraic groups in characteristic 0 which are p -groups. (cid:3) If V is an algebraic variety defined over a valued field K , there is a natural notionof a bounded subset of V ( K ). In [13, pp. 81–83], this is presented for completevalued fields with archimedean value group, but it readily generalizes to arbitraryvalued fields, and the results we will use from [13] hold in the general case as well.If V is an affine variety, a subset B ⊆ V ( K ) is said to be bounded in V ( K ) if forany f ∈ K [ V ] there is γ ∈ Γ K such that val( f ( B )) ⊆ [ γ, ∞ ]. If V is an arbitraryvariety and U , . . . , U m an open affine covering of V , a subset B ⊆ V ( K ) is calledbounded in V ( K ) if for i = 1 , . . . , m there is a set B i ⊆ U i ( K ) which is bounded in U i ( K ) such that B = S i B i . We will use the following properties of bounded sets: One may first reduce the statement to linear algebraic groups. For these, by the Jordandecomposition, the result follows from the unipotent and the semisimple cases, which are clear.
Fact 2.15 ([13, pp. 81–82]) . (1) If f : V → W is a morphism of algebraic varieties and B ⊆ V ( K ) isbounded in V ( K ) , then f ( B ) is bounded in W ( K ) .(2) If V is a closed subvariety of W and B ⊆ V ( K ) , then B is bounded in V ( K ) if and only if B is bounded in W ( K ) .(3) If f : V → W is a proper morphism of quasi-projective algebraic varietiesand B ⊆ W ( K ) is bounded in W ( K ) , then f − ( B ) is bounded in V ( K ) . (cid:3) In what follows, as in [9], boundedness of a definable subset D of V in an algebraicvariety V just means boundedness of D ( K ) in V ( K ), where K | = ACVF and D and V are defined over K . This does not depend on the model K . Lemma 2.16.
Let G be an algebraic group, G a normal algebraic subgroup and G := G/G , with projection map π : G → G . For a definable subgroup H of G ,the following are equivalent:(1) H is bounded in G ;(2) H ∩ G is bounded in G and π ( H ) is bounded in G .Proof. (1) ⇒ (2) follows from Fact 2.15(1+2). To prove the converse, we may assumethat H is Zariski dense in G and that G is a connected algebraic group. Claim.
There is a definable subset W of H which is bounded in G and such that π ( W ) = π ( H ) . To settle the claim, we first note that by the proof of [12, Corollary 4.5] thereis a definable subset Y ⊆ H such that π ( Y ) = π ( H ) and π ↾ Y has finite fibers. Asalgebraic closure (in the field sort) in ACVF is given by field theoretic algebraicclosure, we may assume that Y = H ∩ e Y for some constructible subset e Y of G suchthat π ↾ e Y has finite fibers. Since π ( e Y ) is a Zariski dense constructible subset of G , there is an open affine subvariety U ⊆ G contained in π ( e Y ) such that, setting Z U := π − ( U ) ∩ e Y , the map π : Z U → U is a (surjective and) finite morphism, sois in particular proper. Moreover, note that π ( Z U ∩ H ) = U ∩ π ( H ).As H is Zariski dense in G , we may choose finitely many elements h , . . . , h n ∈ H such that ( U i ) ≤ i ≤ n is an (affine) open cover of G , where U i := π ( h i ) U . Thus π ( h i Z U ∩ H ) = U i ∩ π ( H ) for all i . Since π ( H ) is bounded in G by assumption,there are bounded definable subsets B i of U i such that π ( H ) = S ni =1 B i . As π : h i Z U → U i is proper, W i := π − ( B i ) ∩ h i Z U is bounded in h i Z U by Fact 2.15(3), sobounded in G as well, by Fact 2.15(2). Thus W := S ni =1 W i is as required, provingthe claim.The set ( H ∩ G ) × W is bounded in G × G , and so ( H ∩ G ) · W is bounded in G , as it is the image of a bounded set under the multiplication map m : G → G .Since H ⊆ ( H ∩ G ) · W , boundedness of H in G follows. (cid:3) Proposition 2.17.
Let N be a definable stably dominated subgroup of an algebraicgroup G . Then b N is definably compact.Proof. We may assume that G is a connected algebraic group. We will first showthat N is a bounded subset of G . By Chevalley’s structure theorem, there is asurjective homomorphism of algebraic groups π : G ։ A , with A an abelian varietyand L := ker( π ) a linear algebraic group. By [12, Corollary 4.5], N ∩ L is a stablydominated group, and so it follows from [12, Proposition 6.9] that ( N ∩ L ) is a EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 13 bounded subset of its Zariski closure, which is an algebraic subgroup of L . Thus,( N ∩ L ) and so also N ∩ L is a bounded subset of L . Moreover, π ( N ) is boundedin A , as A is a projective variety and so bounded in itself. Thus N is bounded in G by Lemma 2.16, and N is bounded in G .By Fact 2.11, to prove that b N is definably compact, we may assume that N = N .Let p N be the unique generic type of N . Then the pro-definable set S = { r ( x, y ) ∈ c G | b π i ( r ) = p N , i = 1 , } is closed in c G . Moreover, S ⊆ c N , so S is a bounded subset of c G , in theterminology of [9]. Thus, S is definably compact by [9, Theorem 4.2.19]. Let m denote the multiplication in G . Then Z = b m ( S ) ⊆ b G is a definably compact subsetof b N by [9, Proposition 4.2.9]. The set Z contains all realized types in N ( i.e. ,simple points of b N ), since every element in N is a product of two generics. As thesimple points are dense in b N , we conclude that b N = Z is definably compact. (cid:3) Continuity of the tensor product.
In this subsection, we continue to workin ACVF.
Lemma 2.18.
For any definable sets V , W , the map ⊗ : b V × c W → \ V × W ispro-definable.Proof. Let p ( x ) ∈ b V , q ( y ) ∈ c W and r ( x, y ) = p ( x ) ⊗ q ( y ). Fix some formula φ ( x, y ; t ). Write d p φ ( x ; y, t ) = θ b ( y, t ), where θ depends only on φ and b is thecanonical parameter of θ b , so that b is definable from p seen as a point in b V . Forany parameter c , we have φ ( x, y ; c ) ∈ r ⇐⇒ c | = d q θ b ( y ; t ). This is definable from( p, q ) ∈ b V × c W . (cid:3) Assume H is a finite-dimensional vector space over K | = ACVF. A semi-lattice in H is an O -submodule u ≤ H for which there is a vector subspace U ≤ H suchthat u/U is a lattice in H/U . Clearly, the set of semi-lattices L ( H ) is definable.The linear topology on L ( H ) is the definable topology whose pre-basic open sets areof the form Ω h = { u ∈ L ( H ) | h u } or of the form Θ h = { u ∈ L ( H ) | h ∈ m u } .We leave the easy proof of the following lemma to the reader. (See [9, Chapter 5]for details concerning semi-lattices.). Lemma 2.19. (1) Any semi-lattice u ∈ L ( H ) is isomorphic to O l × K r , where l + r = n := dim( H ) . In particular, there is a basis b of H such that u isdiagonal for b , i.e., u = L ni =1 ( u ∩ Kb i ) .(2) If H = L si =1 H i , then Q si =1 L ( H i ) embeds naturally into L ( H ) . This em-bedding is a definable homeomorphism onto a closed subset of L ( H ) .(3) Let b = ( b , . . . , b n ) be a basis of H . Then the map d : Γ n ∞ → L ( H ) , d ( γ , . . . , γ n ) := ( n X i =1 c i b i | val( c i ) + γ i ≥ for i = 1 , . . . , n ) , is a definable homeomorphism onto the closed subset ∆ b of L ( H ) consistingof all semi-lattices in H which are diagonal for b . In particular, ∆ b is a Γ -internal definable subset of L ( H ) . The following strong result about simultaneous diagonalisation is a converse topart (3) of the preceding lemma.
Fact 2.20 ([9, Lemma 6.2.2]) . Let X ⊆ L ( H ) be a definable Γ -internal subset.Then there are finitely many K -bases b , . . . , b N of H such that any u ∈ X isdiagonal for some b i . Lemma 2.21.
Let H and H ′ be finite-dimensional vector spaces.(1) If u ∈ L ( H ) and u ′ ∈ L ( H ′ ) , then u ⊗ u ′ ∈ L ( H ⊗ H ′ ) , where the tensorproduct is taken over O . Furthermore, the map τ : L ( H ) × L ( H ′ ) → L ( H ⊗ H ′ ) , ( u, u ′ ) u ⊗ u ′ , is definable.(2) Let Σ ⊆ L ( H ) be a Γ -internal definable subset. Then the restriction of τ to Σ × L ( H ′ ) is continuous in the linear topology.Proof. (1) Clear.(2) Let b = ( b , . . . , b n ) be a basis of H . We first show the continuity resultin the special case where Σ = ∆ b ⊆ L ( H ). Let H i = Kb i and E i = H i ⊗ H ′ ≤ H ⊗ H ′ . It follows from Lemma 2.19 that ∆ b = Q ni =1 L ( H i ) ∼ = Γ n ∞ canonically,and that the map τ ↾ ∆ b × L ( H ′ ) decomposes topologically into the product of maps τ i : L ( H i ) × L ( H ′ ) → L ( E i ). The continuity of τ i may be easily checked directly.We leave the verification to the reader. This shows the result for Σ = ∆ b .To prove the general case, using Fact 2.20, we may find bases b , . . . , b N of H such that Σ ⊆ S Ni =1 ∆ b i . Let ( u, u ′ ) ∈ Σ × L ( H ′ ), and let U be an openneighborhood of u ⊗ u ′ in L ( H ⊗ H ′ ). Let I be the set of indices i ∈ { , . . . , N } such that u ∈ ∆ b i . For i ∈ I , choose open neighborhoods Ω i of u and Ω ′ i of u ′ such that τ ((Ω i ∩ ∆ b i ) × Ω ′ i ) ⊆ U . For i / ∈ I , we set Ω i = L ( H ) \ ∆ b i (which is anopen neighborhood of u by Lemma 2.19) and Ω ′ i = L ( H ′ ). Let Ω = T Ni =1 Ω i andΩ ′ = T Ni =1 Ω ′ i . By construction, we get τ ((Ω ∩ Σ) × Ω ′ ) ⊆ U . (cid:3) For n, d ≥ H n,d the vector space of all polynomials of degree ≤ d in n variables over the valued field. In what follows, L ( H n,d ) will be endowed with thelinear topology. For p ∈ c A n let J n,d ( p ) := { h ∈ H n,d | val( h ( p )) ≥ } . Then J n,d ( p )is a definable O -submodule of H n,d , and it is easy to see that it is a semi-lattice. Fact 2.22 ([9, Theorem 5.1.4]) . The maps J n,d : c A n → L ( H n,d ) are pro-definableand continuous, and the induced map J n := ( J n,d ) d : c A n → lim ←− d L ( H n,d ) is apro-definable homeomorphism onto its image. Proposition 2.23.
Let
V, W be algebraic varieties and Σ ⊆ b V an iso-definable Γ -internal subset. Then the map ⊗ : Σ × c W → \ V × W is a (pro-definable) homeo-morphism onto its image.Proof. Clearly, the map ⊗ : Σ × c W → \ V × W is injective and has a continuousinverse, since the inverse is given by the restriction of the canonical map \ V × W → b V × c W . It thus suffices to show that ⊗ : Σ × c W → \ V × W is continuous. Passingto affine charts and some ambient affine spaces, we may assume that V = A n and W = A m .By Fact 2.22, it suffices to show that the map f d = J n + m,d ◦ ⊗ : c A n × d A m → L ( H n + m,d ) restricted to Σ × d A m is continuous for every d . It follows from thedefinitions of the tensor product of generically stable types and of the maps J l,d that f d factors through L ( H n,d ) × L ( H m,d ). More precisely, f d decomposes as c A n × d A m J n,d × J m,d −−−−−−−→ L ( H n,d ) × L ( H m,d ) τ −→ L ( H n,d ⊗ H m,d ) ρ −→ L ( H n + m,d ) , EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 15 where τ is as in Lemma 2.21 and ρ is the (continuous definable) map induced bythe inclusion H n + m,d ⊆ H n,d ⊗ H m,d .As J n,d (Σ) is a Γ-internal definable subset of L ( H n,d ), the continuity of the map f d ↾ Σ × d A m follows from Lemma 2.21. (cid:3) Existence of a definable equivariant retraction
For our construction of an equivariant deformation retraction, we will only needa special case of the main result of [9], namely Fact 3.2 below, which is much easierto prove than the general non-smooth case. Moreover, all deformation retractionswhich will appear in our paper are of a very special form. In order to simplify theexposition, we give them a name.
Definition 3.1.
Let V be an algebraic variety defined over the valued field F , andlet X ⊆ V be an F -definable subset. A pro- F -definable continuous map H : [0 , ∞ ] × b X → b X is called an F -definable special deformation retraction of b X with final image Σ ifthe following properties hold:(i) H ∞ = id b X (ii) H ( b X ) ⊆ Σ (iii) H t ↾ Σ = id Σ for all t ∈ [0 , ∞ ].(iv) For every open subvariety U of V , the set b U ∩ b X is invariant under H .(v) H (0 , x ) = H (0 , H ( t, x )) for any x ∈ b X and any t ∈ [0 , ∞ ].(vi) X is invariant under H .(vii) H ( X ) = H ( b X ) = Σ (viii) Σ is F -definably homeomorphic to a definable subset of Γ w , for some finite F -definable set w .(ix) For any x ∈ b X and any t < ∞ , H ( t, x ) is Zariski generic in X .Here, we denoted by H t the map sending x to H ( t, x ). Remark.
Let H be an F -definable special deformation retraction of b X with finalimage Σ . Then (vi) combined with (vii) and (ix) yield the following:(x) Σ ⊆ X , and every element of Σ is Zariski generic in X . Fact 3.2.
Let V be a smooth irreducible quasi-projective variety over a valued field F and let X ⊆ V be a v -clopen F -definable subset such that b X is definably compactand definably connected.Then there is an F -definable special deformation retraction H : [0 , ∞ ] × b X → b X with final image Σ .Proof. That such a map exists with properties (i-vii) follows from [9, Theorem12.1.1 & Remark 12.2.3]. Moreover, as b X is definably connected, one may achievein addition property (viii), by the discussion in [9, top of p. 178].Finally, let us argue why one may achieve property (ix). It follows from theproof of [9, Theorem 12.1.1] that for any x ∈ b X and any t < ∞ , the type H ( t, x )is Zariski generic in V , as this is true for the inflation homotopy H inf defined in[9, p. 180] and H is constructed as the composition of H inf with finitely manyadditional homotopies which are all Zariski generalizing. (cid:3) Lemma 3.3.
Let V be an irreducible variety defined over a model of ACVF , andlet η : [0 , ∞ ] → b V be a continuous definable path with η ( ∞ ) a Zariski generic typein V . Then η is constant in a neighborhood of ∞ .Proof. Let p = η ( ∞ ). For any open subvariety U of V , there exists γ U < ∞ suchthat η ([ γ U , ∞ ]) ⊆ b U . By the construction of b V , it is enough to show that for anyaffine open subvariety U and any regular function f on U , the continuous definablefunction T : [ γ U , ∞ ] → Γ ∞ , T ( t ) := val( f ∗ ( η ( t ))) is constant in a neighborhood of ∞ . If f is not identically 0 on U , then val( f ∗ ( p )) = ∞ , since p is Zariski generic in U by assumption. The result now follows from quantifier elimination in Γ ∞ . (cid:3) Corollary 3.4.
Let V be a smooth irreducible quasi-projective variety defined overa valued field F and let X ⊆ V be a v -clopen F -definable subset such that b X isdefinably compact and definably connected. Let q , q ∞ ∈ b X ( F ) with q Zariskigeneric in V . Then there is an F -definable continuous path η : [0 , ∞ ] → b X with η (0) = q , η ( ∞ ) = q ∞ and η ( t ) Zariski generic in V for any t < ∞ .Moreover, if q , q ∞ ∈ X , then η may be required to have its image in X .Proof. Choose H : [0 , ∞ ] × b X → b X as in Fact 3.2. Let r t = H ( t, q ) and u t = H ( t, q ∞ ). By Lemma 3.3, there is γ < ∞ such that r t = r γ for all t ≥ γ . Moreover,since b X is definably connected and definably compact, so is Σ = H (0 , b X ). AsΣ is definably homeomorphic to a subspace of Γ w for some finite set w , Σ isdefinably path-connected and in particular there is a continuous definable path s : [ α, β ] → Σ between r and u , for some α < β < ∞ , i.e., parametrized by aninterval of finite length. We may glue these three paths, which yields a definablepath η : [0 , ∞ ] → b X between q and q ∞ as required. (Zariski genericity of η ( t ) for t < ∞ is a consequence of the fact that H satisfies property (ix) in Definition 3.1.)The moreover part follows, as H preserves X and Σ ⊆ X . (cid:3) Lemma 3.5.
Let G be an algebraic group defined over a model of ACVF , and let H = H be a connected definable subgroup of G . Then b H is definably connected.Proof. The group H acts transitively and definably on the (finite) set of definableconnected components of b H . The result follows, looking at stabilizers. (cid:3) Theorem 3.6.
Let G be an algebraic group defined over a valued field F , and let N be a commutative F -definable stably dominated connected subgroup of G . Let p e be the type of the identity in N , and let p N be the generic type of N . Then there isan F -definable path q : [0 , ∞ ] → N such that(i) q ∞ = p e and q = p N ;(ii) for any t < ∞ , q t is the generic type of a connected stably dominateddefinable subgroup N t of N which is Zariski dense in N .Proof. First note that p e is strongly stably dominated, as is p N , by Lemma 2.12, so p e , p N ∈ N ( F ). Furthermore, we may assume that G equals the Zariski closureof N , so in particular G is connected and N is Zariski dense in G , hence N is v -clopen in G , as it is a subgroup with non-empty interior for the valuation topology.Moreover, b N is definably compact by Proposition 2.17 and definably connected byLemma 3.5. We may thus infer from Corollary 3.4 that there is an F -definablepath η : [0 , ∞ ] → N , such that η ∞ = p e and η = p N , with η t strongly stablydominated and Zariski generic in N for any t < ∞ . EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 17
For p, q ∈ b G we denote by p − ∈ b G the image of p under the inversion mapin G , and we set p ∗ q := b m ( p ⊗ q ) ∈ b G , where m denotes the multiplicationmap in G . Now fix n large enough and define q t = η t ∗ η − t ∗ · · · where theproduct contains n instances of η t ∗ η − t . Then the map t q t is continuous byProposition 2.23. Moreover, it follows from [9, Proposition 2.6.12] that q t is stronglystably dominated. By [12, Lemma 5.1], q t is the generic of an ∞ -definable connectedsubgroup N t of N . By Lemma 2.13, N t is in fact definable. Clearly, q ∞ = p e and q = p N , since p e and p N are generic types of connected stably dominated definablesubgroups. Moreover, the type q t , obtained as an alternating sum of η t , is stronglystably dominated and Zariski generic in N for any t < ∞ . (cid:3) Theorem 3.7.
Let G be an algebraic group defined over a valued field F , and let N be a commutative F -definable stably dominated connected subgroup of G .Then there is an F -definable special deformation retraction ρ : [0 , ∞ ] × b N → b N with final image { p N } such that ρ is equivariant under the action of N by multipli-cation and for each t < ∞ , q t = ρ ( t, p e ) is the generic type of a connected stronglystably dominated definable subgroup N t of N which is Zariski dense in N .Proof. Let q : [0 , ∞ ] → N be a definable path as in the conclusion of Theorem 3.6.Define r : [0 , ∞ ] × b N → \ N × N , ( t, a ) q t ⊗ a . Then r is continuous by Proposi-tion 2.23, and so is ρ := b m ◦ r : [0 , ∞ ] × b N → b N . Clearly, ρ is N -equivariant withfinal image { p N } and satisfies all the required properties from Definition 3.1. (cid:3) For the definition of the limit stably dominated subgroup of a pro-definable groupwe refer to [12, Definition 5.6].
Lemma 3.8.
Let G be an algebraic group, G a normal algebraic subgroup and G := G/G , with projection map π : G → G . Suppose that in G the limit stablydominated subgroup exists. Let N ≤ G be this group, and suppose that G/N is Γ -internal. Set N := N ∩ G and N := π ( N ) . Then the following holds:(1) For i = 1 , , N i is the limit stably dominated subgroup of G i , and G i /N i is Γ -internal.(2) If N and N are stably dominated, so is N .Proof. (1) Let N = S t | = q S t , where ( S t ) t | = q is a limit stably dominated family for G in the sense of [12, Definition 5.6]. Then S t ∩ G is stably dominated for any t | = q , by [12, Corollary 4.5]. Moreover, any connected stably dominated subgroupof G is a subgroup of N . Thus, N is the limit stably dominated subgroup of G .The group G /N is Γ-internal, as it embeds into G/N .Clearly, any π ( S t ) is stably dominated and connected. Moreover, since G /N = π ( G/N ) is Γ-internal, any stably dominated connected subgroup H of G is neces-sarily a subgroup of N , as HN /N is stably dominated and Γ-internal. It followsthat N = S t | = q π ( S t ) is the limit stably dominated subgroup of G .(2) follows from [12, Lemma 4.3]. (cid:3) Fact 3.9.
Let S be a semi-abelian variety defined over a valued field F ⊆ K | =ACVF . Then there is an F -definable decomposition → N → S → Λ → , with Λ Γ -internal and N stably dominated, definable and connected. The group N is the unique maximal definable stably dominated and connected subgroup of S .Moreover, if S = A is an abelian variety, Λ is definably compact.Proof. If S = A is an abelian variety, the statement is proven in [12, Corollary 6.19].If S = G nm , then G nm ( O ) is stably dominated and connected with quotient groupΛ ∼ = (Γ , +) n , yielding the case of a torus.Now let S be an arbitrary semi-abelian variety. As S is commutative, by [12,Theorem 5.16], S admits a (unique) limit stably dominated definable subgroup N ,with quotient S/N
Γ-internal. The result then follows from Lemma 3.8, as S is theextension of an abelian variety by a torus. (cid:3) Theorem 3.10.
Let S be a semi-abelian variety defined over F ⊆ K | = ACVF ,and let → N → S → Λ → be the decomposition from Fact 3.9.Then there is an F -definable special deformation retraction ρ : [0 , ∞ ] × b S → b S with final image Σ ⊆ b S such that ρ is equivariant under the action of S bymultiplication and for each t < ∞ , q t = ρ ( t, p e ) is the generic type of a connectedstrongly stably dominated definable subgroup N t of N , with N t Zariski dense in S .Moreover, the morphism π : S → Λ induces definable bijection between Σ and Λ .Proof. We proceed as in the proof of Theorem 3.7. We first define a map r :[0 , ∞ ] × b S → \ S × S , ( t, a ) q t ⊗ a , where q is as in Theorem 3.6. The map ρ := b m ◦ r : [0 , ∞ ] × b S → b S is continuous and satisfies all the required propertiesfrom Definition 3.1, and it is clearly S -equivariant. The restriction ρ S = ρ ↾ [0 , ∞ ] × S has final image equal to { a + p N | a ∈ S } , a set which may be identified with S/N ∼ = Λ and which is thus Γ-internal.It follows from the definitions that ρ is the canonical extension (in the sense of[9, Section 3.8]) of ρ S . Thus, the final image of ρ is equal to Λ as well, as b Λ = Λ. (cid:3) An explicit definable equivariant retraction inequicharacteristic 0
In this section, we will give an alternative, more explicit, construction of anequivariant definable special deformation retraction in the case of equicharacteristic0. This construction does not require knowing in advance that a (non-equivariant)retraction exists. More importantly, it does not require that the stably dominatedconnected group N be commutative, in order to show that its stable completion b N allows for an N -equivariant definable special deformation retraction to the generictype of N . We believe that even in the commutative case, it might be useful in itsown right.4.1. Internality of quotients in ACVF , . The following result will be used todefine an intrinsic scale, given by subgroups, in any stably dominated definablesubgroup of an algebraic group in a model of ACVF , . Lemma 4.1.
Work in
ACVF , . Let D, D ′ be definable subgroups of O . Then thefollowing are equivalent:(1) D ′ ⊆ D .(2) O /D is O /D ′ -internal.(3) O /D is almost O /D ′ -internal. EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 19
Proof. . ⇒ . ⇒ . is clear.We now prove 3 . ⇒ . Note that every definable subgroup of O is of the form γ m or γ O for some 0 ≤ γ ≤ ∞ , so in particular the set of definable subgroups of O is totally ordered by inclusion. Whenever D ( D ′ are definable subgroups, there is0 < γ ∈ Γ such that D ⊆ γ m ( γ O ⊆ D ′ . It is thus enough to show that for any0 < γ ∈ Γ, the set O /γ m is not almost O /γ O -internal.The idea of the proof is similar to [8, Lemma 5.1]. Consider the field of gener-alized power series K = k (( X Γ )), where k is an algebraically closed field of char-acteristic 0 and Γ a non-trivial divisible ordered abelian group. Let l = k (( T Q ))(with the trivial valuation) and consider L = l (( X Γ )) with the X -adic valuation,an elementary extension of K | = ACVF , .Given 0 < γ ∈ Γ( K ) = Γ and ρ ∈ L of valuation 0 (e.g., ρ ∈ k × ) we now definean automorphism σ ρ of L fixing K ∪ ( O /γ O )( L ) pointwise such that for distinct ρ, ρ ′ ∈ k × , we have σ ρ ( T + γ m ) = σ ρ ′ ( T + γ m ). This will show the result, as then( O /γ m )( L ) ( acl( K ∪ ( O /γ O )( L )).Define σ = σ ρ on monomials as follows: σ ( aT q X g ) := aT q X g exp( qρX γ ) . Then extend the map by linearity to generalized series. We check that σ is anautomorphism having the required properties:(1) σ is a continuous automorphism: linearity and continuity follow from the defi-nition. Multiplicativity can be checked on monomials: σ ( a T q X g ) · σ ( a T q X g ) = a a T q + q X g + g exp( q ρX γ ) exp( q ρX γ ) = σ ( a a T q + q X g + g ).For any x ∈ O , we have(4.1) v ( x − σ ( x )) ≥ v ( x ) + γ. This, along with continuity, implies that σ is a bijection: if we have y such that v ( σ ( y ) − x ) =: η , we let y ′ = y + x − σ ( y ) and obtain a better candidate for apreimage of x , that is v ( σ ( y ′ ) − x ) > v ( σ ( y ) − x ). By transfinite induction, we findan actual preimage of x .(2) σ fixes K ∪ ( O /γ O )( L ) pointwise: For x ∈ K , σ ( x ) = x is immediate fromthe definition and (4.1) gives the rest of the statement.(3) σ ρ ( T + γ m ) = σ ρ ′ ( T + γ m ) for ρ = ρ ′ follows from the computation v ( σ ρ ( T ) − σ ρ ′ ( T )) = v (( ρ − ρ ′ ) T X γ ) = γ . (cid:3) Remark 4.2.
In positive and in mixed characteristic, the statement of Lemma 4.1does not hold: in ACVF p,p , given any γ ∈ Γ > and c ∈ O with v ( c ) = γ , theFrobenius automorphism induces a group isomorphism O /γ O = O /c O ∼ = O /c p O = O / ( pγ ) O ; in ACVF ,p , setting γ := val( p ) ∈ Γ > , the map x x p induces adefinable surjection O /γ O ։ O / (2 γ ) O . Corollary 4.3.
Work in
ACVF , . Let D be a definable subgroup of O , and let d ∈ N . Then ( O /D ) d is the maximal (almost) O /D -internal quotient of O d .Proof. It is clear that ( O /D ) d is O /D -internal. Conversely, let O d /N be (almost) O /D -internal for some ( ∞ -)definable subgroup N ≤ O d . For i ∈ { , . . . d } , let B i = { } i − × O × { } d − i − ∼ = O . It follows that for any i the group B i / ( N ∩ B i )is (almost) O /D -internal, and so { } i − × D × { } d − i − ≤ N ∩ B i by Lemma 4.1.Thus D d ≤ N . (cid:3) Corollary 4.4.
Work in
ACVF , . Let C be a model (or more generally a basestructure consisting of a field). Let γ ∈ Γ , and set C γ := acl( Cγ ) . Let D be a C γ -definable subgroup of O . If a = ( a , . . . , a d ) is generic in O d over C , the tuple ( a /D, . . . , a d /D ) dcl -generates dcl( C γ a ) ∩ Int C γ ( O /D ) over C γ .More generally, if a ∈ K m realizes a strongly stably dominated type over C ,there is a tuple b = ( b , . . . , b d ) from C ( a ) such that b is generic in O d over C and acl( C γ a ) ∩ Int C γ ( O /D ) is finitely acl -generated by ( b /D, . . . , b d /D ) over C γ .Proof. First suppose that a is generic in O d over C . By the previous corollary,( O /D ) d is the maximal (almost) O /D -internal quotient of O d , over any set ofparameters. As tp( a/C ) is stably dominated and thus orthogonal to Γ, the tuple a is generic in O d over C γ . By the last sentence of Proposition 2.9, dcl( aC γ ) ∩ Int C γ ( O /D ) is then interdefinable with ( a /D, . . . , a d /D ) over C γ , proving theresult.Now suppose a ∈ K m with tp( a/C ) strongly stably dominated, i.e., tr( a/C ) =tr( k C ( a ) /k C ) = d by Fact 2.5. We find b ∈ C ( a ) generic in O d over C . As b and a areinteralgebraic over C , we have acl( Cγb ) ∩ Int Cγ ( O /D ) = acl( Cγa ) ∩ Int Cγ ( O /D ),so the result follows from the special case. (cid:3) Linearization.
In this subsection, we work in ACVF , . Let C be a basestructure consisting of a field. We fix a C -definable stably dominated connectedsubgroup N of an algebraic group. For γ ∈ Γ, let C γ = acl( Cγ ). Set d := dim( N ). Fact 4.5 ([8, Section 7]) . For γ ∈ Γ , O /γ O and O /γ m are stably embedded. By Proposition 2.9, we let N γ be the kernel of the maximal O /γ O -internalquotient of N and similarly N + γ is defined as the kernel of the maximal O /γ m -internal quotient of N , both computed over C γ .The groups N γ and N + γ are ∞ -definable. Note that if γ < δ , then N γ ⊇ N + γ ⊇ N δ ⊇ N + δ . Lemma 4.6.
The groups N γ and N + γ have bounded index inside some definablegroup and are intersections of definable groups.Proof. By Fact 2.6(1), the quotient
N/N γ can be written as an inverse limit ofdefinable groups π i ( N ). By Lemma 2.12, the generic type p N of N is stronglystably dominated and by Corollary 4.4, over C γ , the set acl( C γ a ) ∩ Int C γ ( O /γ O ) isin the algebraic closure of a finite set in O /γ O . By construction of N γ , if a | = p N | C ,then acl( C γ a ) ∩ Int C γ ( O /γ O ) is interalgebraic over C γ with the sequence π i ( a ).Hence we can find i such that acl( C γ a ) ∩ Int C γ ( O /γ O ) is already interalgebraicwith π i ( a ) over C γ . Then N γ has bounded index inside the kernel of π i . Also N γ is the intersection of the kernels of π i which are definable subgroups of N .The same arguments work for N + γ . (cid:3) Lemma 4.7.
The group N γ is strongly stably dominated.Proof. We first show that the quotient g γ := N γ /N + γ is stable of Morley rank d .Let p = p N be the generic type of N , and let Lin( p ) be the generic type of O d .Then any a | = p | C is interalgebraic over C with some b = ( b , . . . , b d ) | = Lin( p ) | C .Denote by r γ (by r + γ , respectively) the generic type of the maximal O /γ O -internal(maximal O /γ m -internal, respectively) quotient of N over C , with realizations a γ and a + γ , images of a via the canonical projection. Let b γ = ( b /γ O , . . . , b d /γ O ),and define similarly b + γ . We have the following: EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 21 • acl( C γ b + γ ) = acl( C γ a + γ ); • acl( C γ b γ ) = acl( C γ a γ ) and as tp( b + γ / acl( C γ b γ )) is interdefinable with thegeneric of a k d -torsor, then tp( a + γ / acl( C γ a γ )) is interalgebraic with thegeneric of a k d -torsor. In particular, that type is stable of Morley rank d . • The generic of ker(
N/N + γ → N/N γ ) ∼ = g γ is interdefinable with the non-forking extension of tp( a + γ /C γ a γ ) by Lemma 2.8, hence g γ is a stable groupof Morley rank d .Any type p γ of N γ projecting on the generic of g γ is strongly stably dominated. Inparticular, it is definable over C γ . The same is true for any translate of p γ , sincethe generic of g γ is invariant under translation. Hence p γ is a generic type of N γ and N γ is strongly stably dominated. (cid:3) Corollary 4.8.
The groups N γ and N + γ are definable.Proof. By Lemma 4.7, N γ is strongly stably dominated, so it is definable byLemma 2.13.Now by the proof of Lemma 4.7, the quotient N γ /N + γ is ω -stable. By Lemma 4.6,write N + γ = T i H i , where the H i ’s are definable subgroups of N γ . Since there is noinfinite descending chain of definable subgroups of N γ /N + γ , the intersection T i H i is equal to a finite subintersection, hence N + γ is definable. (cid:3) Corollary 4.9.
The group N γ = N γ is of finite index in N γ .Proof. This is just Fact 2.11, combined with Fact 2.6(2). (cid:3)
Lemma 4.10.
Let γ ∈ [0 , ∞ ] . Set V γ = S δ>γ N δ and W γ = T δ<γ N δ . Then thefollowing holds:(1) V γ = S δ>γ N + δ ≤ N + γ and ( N + γ : V γ ) < ∞ , so in particular (cid:0) N + γ (cid:1) = V γ .(2) N γ ≤ W γ = T δ<γ N + δ and ( W γ : N γ ) < ∞ , so in particular N γ = W γ .Proof. We know that V γ = S δ>γ N + δ ⊆ N + γ and N γ ⊆ W γ = T δ<γ N δ = T δ<γ N + δ .Let us show that N γ = W γ . We fix some saturated model M and we work over C γ . Suppose that the index of N γ in W γ is infinite. Let a ∈ M realize the genericof N over C γ and let a γ be the image of a in N/N γ under the projection map, asin Lemma 4.7, and let ˜ a γ be the image of a in N/W γ . We have that ˜ a γ is algebraicover a γ and that a δ , δ < γ is algebraic over ˜ a γ . However, by assumption, a γ isnot algebraic over ˜ a γ and there exists an automorphism fixing C γ ˜ a γ along with a δ , δ < γ and for which a γ has an infinite orbit. We now define b δ , δ ≤ γ as in the proofof Lemma 4.7. Then each b δ is interalgebraic with a δ . Therefore b γ has an infiniteorbit under σ whereas each b δ , δ < γ has a finite orbit. Since b δ ∈ dcl( C γ b δ ′ ) for δ < δ ′ , taking a large enough power of σ , we may assume that σ actually fixes each b δ , δ < γ . This is impossible since any element d of O /γ O is determined by thesequence ( d δ : δ < γ ) where d δ is the image of d in O /δ O .The fact that ( N + γ : V γ ) < ∞ is proved in a similar way. Assume that V γ hasinfinite index in N + γ . Let ˜ a + γ be the image of a in N/V γ , and a + γ that in N/N + γ .Take as above an automorphism σ fixing C γ a + γ and under which ˜ a + γ has infiniteorbit. This implies that each a δ , δ > γ has infinite orbit. The same is true foreach b δ , δ > γ , however b + γ has finite orbit under σ . Taking a power of σ , we mayassume that b + γ is fixed. As b + γ = b + γ m , using σ ( γ ) = γ , we get σ ( b ) + γ m = σ ( b + γ ) and so δ = v ( b − σ ( b )) > γ . This implies that σ ( b δ ) = b δ , which is the desiredcontradiction. (cid:3) Corollary 4.11.
For any γ ∈ [0 , ∞ ] , one has (cid:0) N + γ (cid:1) = (cid:0) N + γ (cid:1) .Proof. This follows from S δ>γ N δ = (cid:0) N + γ (cid:1) and Corollary 4.9 since an increasingunion of strongly connected groups is strongly connected. (cid:3) Lemma 4.12.
Let H be a definable stably dominated group such that N γ ⊇ H ⊇ (cid:0) N + γ (cid:1) . Then ( N γ : H ) < ∞ .Proof. We work over a model M containing C γ . Suppose acl( a γ ) ( acl( a ′ ) ⊆ acl( a + γ ), where, a γ , a ′ and a + γ are the images of a | = p N | M under the canonicalprojection of N onto N/N γ , N/H and
N/N + γ , respectively.As non-forking extensions/restrictions and translates of (strongly) stably dom-inated types are (strongly) stably dominated, it follows from Lemma 2.8 thattp( a/a γ ) is generic in a + N γ and strongly stably dominated. Similarly, tp( a/a ′ ) isgeneric in a + H and stably dominated.As a + γ + N + γ ∈ St a γ ( a ) is of Morley rank d over a γ , it follows that St a γ ( a ) isinteralgebraic over a γ with a + γ . Thus there is some non-algebraic element fromSt a γ ( a ) which is in acl( a ′ ). Note that St a γ ( a ) and St a ′ ( a ) are interdefinable over a ′ ,since a ′ ∈ St a γ ( a ) and a γ ∈ dcl( a ′ ) ⊆ dcl( a ) (cf. [7, Lemma 3.5 & Remark 7.9]). Itfollows that dim(St a ′ ( a )) < d , and so tp( a/ acl( a ′ )) is not strongly stably dominatedby Fact 2.5. Thus the generic type of H is not strongly stably dominated either,contradicting Lemma 2.12. (cid:3) Remark 4.13.
Any ∞ -definable group H such that N γ ⊇ H ⊇ N + γ is definable,since H/N + γ is definable, as it is an ∞ -definable subgroup of an ω -stable group.4.3. Definability and continuity.
We keep the notation and conventions fromSubsection 4.2.
Lemma 4.14.
There are C -definable families (cid:16) e N γ (cid:17) γ ∈ [0 , ∞ ] and (cid:16) e N + γ (cid:17) γ ∈ [0 , ∞ ] ofdefinable sugroups of N such that for any γ , N γ ≤ e N γ with ( e N γ : N γ ) < ∞ andsimilarly N + γ ≤ e N + γ with ( e N + γ : N + γ ) < ∞ .In addition, we may choose these families so that e N + γ ≤ e N γ ≤ e N + δ ≤ e N δ when-ever γ < δ .Proof. For γ ∈ [0 , ∞ ], there is a formula ψ ( x, y ) with parameters from C suchthat ψ ( x, γ ) defines N γ . Corollary 4.4 together with Proposition 2.9 implies thatthere are a generic in N/N γ over C γ and b generic in ( O /γ O ) d over C γ such that b ∈ dcl( C γ a ) and a ∈ acl( C γ b ). By compactness, there is θ ( y ) ∈ tp( γ/C ) such thatwhenever | = θ ( δ ), then ψ ( x, δ ) defines a subgroup N ′ δ of N such that there are a generic in N/N ′ δ over C γ and b generic in ( O /γ O ) d over C γ with b ∈ dcl( C γ a ) and a ∈ acl( C γ b ). It follows from Proposition 2.9 that N δ ≤ N ′ δ and ( N ′ δ : N δ ) < ∞ forany such δ . By compactness, we obtain a C -definable family (cid:16) e N γ (cid:17) γ ∈ [0 , ∞ ] with therequired properties.In exactly the same way, one proves the existence of the family (cid:16) e N + γ (cid:17) γ ∈ [0 , ∞ ] .In order to achieve the additional requirement, it is enough to replace the group e N γ by T γ ≤ δ e N δ ∩ T γ<δ e N + δ , and similarly e N + γ by T γ ≤ δ e N δ ∩ T γ ≤ δ e N + δ . (cid:3) EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 23
Lemma 4.15.
The theory
ACVF admits elimination of ∃ ∞ in imaginary sorts.Proof. By [6, Lemma 2.6.2], if D is a definable (imaginary) set, then either it is k -internal, or, for some m , there is a definable surjective map from D m to an infiniteinterval of Γ. The second case is an open condition and implies that D is infinite. Inthe first case, since we know that k eliminates ∃ ∞ , we also see that D being infinitecan be expressed as a definable condition on the parameters defining D . (cid:3) Lemma 4.16.
Let q γ ∈ b N be the generic type of N γ . Then the map q : [0 , ∞ ] → b N , γ q γ is pro-definable. Its image is iso-definable and Γ -internal.Proof. Let (cid:16) e N γ (cid:17) γ ∈ [0 , ∞ ] be a family as given by Lemma 4.14. Then N γ = e N γ forall γ . For every formula φ ( x, y ), the setGen( φ ) := { ( γ, p ) ∈ [0 , ∞ ] × b N | p is φ -generic in e N γ } is relatively definable in [0 , ∞ ] × b N . This can be seen as follows. By generic stability,the φ -definitions of elements of b N are uniform. Now e N γ acts on the φ -definitionsof elements of ce N γ . By Lemma 4.15, there is n φ such that whenever the orbit ofa φ -definition under this action is finite, it is of cardinality at most n φ . It followsthat the set Gen := T φ Gen( φ ) ⊆ [0 , ∞ ] × b N is pro-definable.The set PrGen := { ( γ, p ) ∈ Gen | p ∗ p = p } is then pro-definable as well, and itis equal to Graph( q ), since for a generic type p of a stably dominated group H , onehas p ∗ p = p precisely if p is the principal generic of H . This proves pro-definabilityof q .As q is injective, iso-definability of its image follows from compactness and thenthis image is Γ-internal by definition. (cid:3) Theorem 4.17.
The map q is continuous. Thus, q : [0 , ∞ ] → N is a definablepath between q ∞ = p e and q = p N along generic types of strictly increasing stronglystably dominated connected definable subgroups of N , with q γ Zariski generic in N for every γ < ∞ .Proof. By definable compactness of c N γ (Proposition 2.17), l + γ := lim δ | = γ + q δ existsand is in c N γ . We claim that l + γ = q γ . To see this, first note that, for any δ > γ and a ∈ N δ , we have a · q δ = q δ , hence also a · l + γ = l + γ , as the map q a · q iscontinuous (e.g., by Proposition 2.23). Moreover, as q δ ∗ q δ = q δ for all δ , we have l + γ = l + γ ∗ l + γ by continuity of ⊗ (Proposition 2.23). Hence l + γ is the generic type ofa subgroup of N γ containing S δ>γ N δ = N + γ (Lemma 4.10). By Lemma 4.12, l + γ must be the generic of N γ , namely l + γ = q γ as required.We next show continuity at γ − (including γ = ∞ ): As before, by continuity of ⊗ ,we know that l − γ := lim δ | = γ − q δ is an idempotent. Moreover, since q γ ∗ q δ = q δ , forevery δ < γ , this holds in the limit as well, i.e. , q γ ∗ l − γ = l − γ . As l − γ ∈ T δ<γ c N δ = d W γ ,it follows from Lemma 4.10 that l − γ = q γ , since the only idempotent generic typeof W γ is the principal generic. (cid:3) The proofs of Theorem 3.7 and Theorem 3.10, respectively, show that Theo-rem 4.17 yields the following two corollaries.
Corollary 4.18.
Let G be an algebraic group defined over a model of ACVF , ,and let N be a C -definable stably dominated connected subgroup of G . Then there is a C -definable special deformation retraction ρ : [0 , ∞ ] × b N → b N with final image ρ (0 , b N ) = { p N } such that ρ is equivariant under the action of N by multiplication and for each t < ∞ , q t = ρ ( t, p e ) is the generic type of a definablestrongly stably dominated connected Zariski dense subgroup N t of N , with N s ) N t if s < t . Corollary 4.19.
Let S be a semi-abelian variety defined over F ⊆ K | = ACVF , ,and let → N → S → Λ → be the decomposition from Fact 3.9.Then there is an F -definable special deformation retraction ρ : [0 , ∞ ] × b S → b S with final image Σ := ρ (0 , b S ) ⊆ b S such that ρ is equivariant under the action of S by multiplication and for each t < ∞ , q t = ρ ( t, p e ) is the generic type of a definablestrongly stably dominated connected subgroup N t of N which is Zariski dense in S and such that N s ) N t whenever s < t . Moreover, Σ is in definable bijection with Λ , canonically. Application to the topology of S an Let S be a semi-abelian variety defined over a valued field F with Γ F ≤ R . Inthis section, using the methods and results from [9, Chapter 14], we will deducefrom our results on b S the existence of an equivariant strong deformation retractionof (the underlying topological space of) the Berkovich analytification S an onto itsskeleton, which is a simplicial complex carrying the structure of a piecewise lineargroup.Recall that if B is a parameter set in some model of ACVF, a type p ∈ S ( B )is almost orthogonal to Γ, denoted by p ⊥ a Γ, if Γ( Ba ) = Γ( B ) for a | = p . If thisis the case then for a B -definable function g : X → Γ ∞ , with p concentrating onthe definable set X , we set g ( p ) := g ( a ), where a | = p . This is well-defined, as thevalue g ( a ) does not depend on the realization a .Until the end of this section, we fix a valued field F with Γ F ≤ R . We will nowintroduce some objects and notation (mostly) from [9, Chapter 14]: • R ∞ denotes R ∪ {∞} , equipped with the order topology. Moreover, R + ∞ := R + ∪ {∞} denotes the sub-interval of its non-negative elements. • We set F := ( F, R ). In particular, R ∞ = Γ ∞ ( F ) and R + ∞ = [0 , ∞ ]( F ). • Let V be an algebraic variety defined over F and X ⊆ V × Γ n ∞ an F -definable subset. As a set, let B F ( X ) := { p ∈ S X ( F ) | p ⊥ a Γ } . • We endow B F ( V ) with the topology whose basic open sets are given byfinite intersections of sets of the form { p ∈ B F ( U ) | g ( p ) ∈ Ω } , where U isan open affine subvariety of V defined over F , Ω ⊆ R ∞ is an open intervaland g = val ◦ G for some G ∈ F [ U ]. • We endow B F ( V × Γ n ∞ ) = B F ( V ) × R n ∞ with the product topology, andfinally B F ( X ) ⊆ B F ( V ) × R n ∞ with the subspace topology. • Any F -definable map f : X → Y induces a map B F ( f ) : B F ( X ) → B F ( Y ),which is continuous in case f is the restriction of a regular map betweenthe ambient algebraic varieties. • We fix a maximally complete algebraically closed extension F max of F ,with k F max = ( k F ) alg and Γ F max = R . (Note that such an F max is uniquelydetermined up to F -isomorphism.)The following is an adaptation of Definition 3.1 to the Berkovich setting. EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 25
Definition 5.1.
Let V be an algebraic variety defined over the valued field F , andlet X ⊆ V be an F -definable subset. A continuous map e H : R + ∞ × B F ( X ) → B F ( X )is called a special deformation retraction of B F ( X ) with final image Z if the follow-ing properties hold: • e H ∞ = id B F ( X ) • e H ( B F ( X )) ⊆ Z • e H t ↾ Z = id Z for all t ∈ R + ∞ . • For every open subvariety U of V defined over F , the set B F ( U ) ∩ B F ( X )is invariant under e H . • e H (0 , x ) = e H (0 , e H ( t, x )) for any x ∈ B F ( X ) and any t ∈ R + ∞ . • Z is homeomorphic to a piecewise linear subset of R n , for some n ∈ N . • For any x ∈ B F ( X ) and any t < ∞ , e H ( t, x ) is Zariski generic in X . Fact 5.2.
Let
V, W be algebraic varieties defined over F , and let X ⊆ V × Γ n ∞ and Y ⊆ W × Γ m ∞ be F -definable subsets.(1) The space B F ( V ) is canonically homeomorphic to the underlying topologicalspace of the Berkovich analytification V an of V .(2) The restriction map π X : b X ( F max ) → B F ( X ) is surjective, continuous andclosed. If F = F max , it is a homeomorphism.(3) Any continuous pro- F -definable function h : b X → b Y induces a continuousfunction ˜ h : B F ( X ) → B F ( Y ) such that π Y ◦ h = ˜ h ◦ π X .(4) Let H : [0 , ∞ ] × b X → b X be an F -definable special deformation retraction,with final image H (0 , b X ) = Z . Let Z = π X ( Z ( F max )) . Then H induces aspecial deformation retraction e H : R + ∞ × B F ( X ) → B F ( X ) with final image Z .Proof. This is a combination of 14.1.1, 14.1.2, 14.1.3 and 14.1.6 in [9], except forthe fact in (4) that e H is a special deformation retraction when H is assumed to bean F -definable special deformation retraction. This is easily checked by hand. (cid:3) We will call a map ˜ h : B F ( X ) → B F ( Y ) as in part (3) of Fact 5.2 definablyinduced . Theorem 5.3.
Let G be an algebraic group defined over F , and let N be an F -definable stably dominated connected subgroup of G . Assume that N is commutativeor that F is a valued field of equicharacteristic 0. Then there is a definably induced N ( F ) -equivariant special deformation retraction ˜ ρ : R + ∞ × B F ( N ) → B F ( N ) withfinal image ˜ ρ (0 , B F ( N )) = { p N | F } .Proof. It suffices to apply Fact 5.2(4) to Theorem 3.7 or to Corollary 4.18, respec-tively. (cid:3)
Similarly, Fact 5.2(4) applied to Theorem 3.10 or to Corollary 4.19, respectively,yields the following result, where we identify (the underlying topological space of) S an with B F ( S ). Theorem 5.4.
Let S be a semi-abelian variety defined over F . Consider the F -definable decomposition → N → S → Λ → from Fact 3.9. Then there is adefinably induced S ( F ) -equivariant special deformation retraction ˜ ρ : R + ∞ × S an → S an with final image a skeleton Σ . The map π : S → Λ induces a bijection between Σ and Λ( F ) , where Λ( F ) is a subset of R n carrying the structure of a piecewiselinear abelian group. (cid:3) Remark 5.5.
Recall that in any NIP theory, in particular in ACVF, every ∞ -definable group G admits a strong connected component G , by a result of Shelah(see, e.g., [15, Theorem 8.4]). Assume now that S is a semi-abelian variety definedover a model of ACVF, and let 0 → N → S → Λ → N = N , we have S/S ∼ = Λ / Λ . Thus the failure of strongconnectedness of S may be entirely pushed to the piecewise linear world.Moreover, if S = A is an abelian variety, in the expansion of Γ to a real closedfield R , we get Λ ∼ = S ( R ) e for some integer e with 0 ≤ e ≤ dim( A ), and A an ishomotopy equivalent to S ( R ) e ∼ = Λ / Λ . In this sense, the homotopy type of A an is encoded in the failure of strong connectedness of A .6. NIP abelian groups
In this section we prove an analogue of our main theorem, but starting with anarbitrary abelian group definable in an NIP theory. The result is of course muchweaker, in particular there is no space of generics that could play the role of b V .We obtain a directed system of ∞ -definable subgroups C ( t ) and instead of thosegroups having a stably dominated (or generically stable) generic type—which neednot exist in general—we ask that they admit a generically stable invariant measure .Groups with such a generically stable invariant measure are called fsg and can bethought of as the definably compact groups in a general NIP theory.We first recall some definitions, all of which can be found in more details in [15].A (Keisler) measure µ on a definable set X over a model M is a finitely additiveprobability measure on M -definable subsets of X . If X = G is a group, then wesay that µ is invariant if µ ( g · Y ) = µ ( Y ) for every M -definable Y ⊆ X and every g ∈ G ( M ). A group admitting an invariant measure (over some, or equivalentlyany, model) is said to be definably amenable. In particular, if G ( M ) is amenableas a pure group, then it is definably amenable and it follows that any solvabledefinable group is definably amenable.Since a type is a special case of a measure (with values in { , } instead of[0 , a , . . . , a n aretuples from a model M and φ ( x ) is a formula over M , we let Av ( φ ( x ); a , . . . , a n )denote |{ i ≤ n : M | = φ ( a i ) }| . Definition 6.1.
Let T be an NIP theory and µ ( x ) a measure over a model M .We say that µ is generically stable if it can be uniformly approximated by finiteaverages of points in the following sense:For any formula φ ( x ; y ) and ǫ >
0, there are a , . . . , a n ∈ M | x | such that for any b ∈ M | y | , | µ ( φ ( x ; b )) − Av ( φ ( x ; b ); a , . . . , a n ) | ≤ ǫ. Note that a generically stable measure µ ( x ) is definable in the following sense:given φ ( x ; y ) and ǫ >
0, there are formulas θ i ( y, ¯ d ), i = 1 , . . . , n , that cover y -spaceand such that for any b, b ′ if for some i ≤ n , b, b | = θ i ( y, ¯ d ), then | µ ( φ ( x ; b )) − EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 27 µ ( φ ( x ; b ′ )) | ≤ ǫ . Indeed, one can take ¯ d to consist of the points a , . . . , a n as in thedefinition above for ǫ/ θ i ( y, ¯ d ) enumerate the φ -types over ¯ d .Note in particular that as φ ( x ; y ) and ǫ vary, we only need | T | -many parametersfor ¯ d , since it is enough to take ǫ rational.An ∞ -definable group G which admits a generically stable invariant measure µ ( x ) is said to have fsg ( finitely satisfiable generics ). This condition can be thoughtof as an abstract version of compactness. For instance, in an o-minimal structure,a definable group has fsg if and only if it is definably compact. Similarly we canstate: Lemma 6.2.
Let A be an abelian variety definable in a model of ACVF , then A has fsg.Proof. Let 0 → N → A → Λ → N is stably dominated (hence generically stable), so it has fsg. By [12, Corollary 6.19],Λ is a definably compact, Γ-internal group, hence Λ has fsg by [10, Theorem 8.1].By Proposition 4.5 of that same paper, an extension of a group with fsg by a groupwith fsg has fsg. Hence A has fsg. (cid:3) If µ ( x ) and η ( x ) are two generically stable (Keisler) measures on a definablegroup G , we can define the convolution µ ∗ η ( x ) by µ ∗ η ( φ ( x )) = Z y µ ( φ ( x · y )) dλ ( y ) . We refer to [15, Section 7.4] for explanation of why this integral makes sense. Then µ ∗ η is again a generically stable measure. Note that if G is abelian, then for any g ∈ G ,(A) g · ( µ ∗ η ) = ( g · µ ) ∗ η = µ ∗ ( g · η ) . Proposition 6.3.
Let T be NIP and let G be a definable abelian group. Thenthere is a pro-definable set S and ∞ -definable subgroups N ( t ) , for t ∈ S forming adirected system (any small family has an upper bound) and such that S t ∈ S N ( t ) = G and each N ( t ) stabilizes a generically stable measure on G .Proof. As G is abelian, it is definably amenable. Let M be a | T | + -saturated modeland let µ M be a G -invariant measure over M . We can extend µ M to a globalmeasure µ which is G ( M )-invariant and generically stable (see [11, Lemma 7.6] or[14, Proposition 3.4]). Let N µ = { g ∈ G ( U ) | g · µ = µ } be the stabilizer of µ .Then N µ is a subgroup of G containing G ( M ). By definability of µ , N µ is ∞ -definable over a set of size | T | . Let t be an enumeration of the parameters neededto define µ and write µ = µ t . Then also N µ is defined over t and we can write N µ = N ( t ). Let S = tp( t ). Then S is pro-definable and for every tuple t ∈ S , µ t is a well-defined generically stable measure (where µ t is defined over t using thesame definition scheme as µ over t ). Then also N µ t = N ( t ) is defined over t thesame way N µ is defined over t . We have constructed a pro-definable family of ∞ -definable subgroups of G .Since N ( t ) contains G ( M ), by compactness, for any (small) model M ′ , there is t ′ ≡ t such that N ( t ′ ) contains G ( M ′ ). This proves that S t ∈ S N ( t ) = G .It remains to show that the family is directed. Let ( r i ) i ∈ I be a small family ofelements of S and write µ i = µ r i . For any finite I ⊆ I , let µ I = ∗ i ∈ I µ i . As G isabelian, this product is independent of the order of the factors. By (A), each N ( r i ), i ∈ I is in Stab ( µ I ). Fix a model M ′ over which all the µ i ’s are defined. Let µ be a limit of the µ I over M ′ along an ultrafilter containing { I : I ⊇ I } for eachfinite I ⊆ I . Then µ is M ′ -invariant and its stabilizer contains each N ( r i ), i ∈ I .As above, we can extend µ to a global measure λ which is generically stable andwhose stabilizer still contains all the N ( r i )’s. Now λ is defined over some model ofsize | T | and hence there is an automorphism σ such that σ ( λ ) =: λ is genericallystable over M . Let t be as in the first paragraph of the proof. Then N ( t ) contains G ( M ) and as λ is finitely satisfiable in M , we have λ ( N ( t )) = 1. This impliesthat Stab ( λ ) ≤ N ( t ). So Stab ( λ ) ≤ N ( σ − ( t )), with σ − ( t ) ∈ S . Thereforeeach N ( t i ) is a subgroup of N ( σ − ( t )), which finishes the proof. (cid:3) Corollary 6.4. If G is an abelian group with no indiscernible linearly orderedfamily of ∞ -definable groups, then G has fsg.Proof. By the previous proposition and since the family C ( t ) is directed, it mustbe that G = C ( t ) for some t . Then G stabilizes a generically stable measure andhence G has fsg. (cid:3) References [1] Hans Adler, Enrique Casanovas, and Anand Pillay. Generic stability and stability.
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EFINABLE EQUIVARIANT RETRACTIONS IN NON-ARCHIMEDEAN GEOMETRY 29 [15] Pierre Simon.
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Institut f¨ur Mathematische Logik und Grundlagenforschung, Westf¨alische Wilhelms-Universit¨at M¨unster, Einsteinstr. 62, D-48149 M¨unster, Germany
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