SIMPLE ALMOST HYPERDEFINABLE GROUPS

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SIMPLE ALMOST HYPERDEFINABLE GROUPS ITAI BEN-YAACOV Abstract. (i) We lay down the groundwork for the treatment of almost hyperde- finable groups: notions from [BTW04] are put into a natural hierarchy, and new notions, essential to the study to such groups, fit elegantly into this hierarchy. (ii) We show that “classical” properties of definable and hyperdefinable groups in simple theories can be generalised to this context. In particular, we prove the existence of stabilisers of Lascar strong types and of the connected and locally connected components of subgroups, and that in a simple one-based theory an almost hyperdefinable group is bounded-by-abelian-by-bounded. Introduction This paper is concerned with the generalisation of results from [Wag05, Wag01] to the context of ?/?-groups (see below for the definition), first introduced as almost hyperde- finable groups in [BTW04]. Loosely speaking, an ?/?-group is a group whose underlying set of elements is of the form G = Gb/R, where Gb is a type-definable set, and R = ?i?I Ri is an equivalence relation which is not type-definable but is only an infinite disjunction of type-definable relations (satisfying some additional properties). There are two aspects to our task. The first is to lay the groundwork for the model- theoretic treatment of such groups.

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Well-definition

SIMPLEALMOSTHYPERDEFINABLEGROUPSITAI¨BEN-YAACOVAbstract.(i)Welaydownthegroundworkforthetreatmentofalmosthyperde-ﬁnablegroups:notionsfrom[BTW04]areputintoanaturalhierarchy,andnewnotions,essentialtothestudytosuchgroups,ﬁtelegantlyintothishierarchy.(ii)Weshowthat“classical”propertiesofdeﬁnableandhyperdeﬁnablegroupsinsimpletheoriescanbegeneralisedtothiscontext.Inparticular,weprovetheexistenceofstabilisersofLascarstrongtypesandoftheconnectedandlocallyconnectedcomponentsofsubgroups,andthatinasimpleone-basedtheoryanalmosthyperdeﬁnablegroupisbounded-by-abelian-by-bounded.IntroductionThispaperisconcernedwiththegeneralisationofresultsfrom[Wag05,Wag01]tothecontextofα/β-groups(seebelowforthedeﬁnition),ﬁrstintroducedasalmosthyperde-ﬁnablegroupsin[BTW04].Looselyspeaking,anα/β-groupisagroupwhoseundWerlyingsetofelementsisoftheformG=Gb/R,whereGbisatype-deﬁnableset,andR=i∈IRiisanequivalencerelationwhichisnottype-deﬁnablebutisonlyaninﬁnitedisjunctionoftype-deﬁnablerelations(satisfyingsomeadditionalproperties).Therearetwoaspectstoourtask.Theﬁrstistolaythegroundworkforthemodel-theoretictreatmentofsuchgroups.Thiswaspartiallydonein[BTW04],wheresomebasicdeﬁnitionsweregivenandtheexistenceofwell-behavedstratiﬁedlocalrankswasprovedforalmosthyperdeﬁnablegroups(andpolygroups).However,inordertostudyagroupwemustconsideritssubgroups,andinthatrespectpreviousworkleavesmuchtowant.Asweconsidersubgroupsofα/β-groupsinthecurrentpaperweﬁndourselvesforcedtoconsiderthenewnotionofβ/β-subgroups,namelysubgroupsHb/R⊆Gb/RwhereHbisnottype-deﬁnable,butagainonlyaninﬁniteunionoftype-deﬁnablesets(withsomeadditionalproperties).Whiledoingsoweﬁndourselvesworkinginaratherweirdcategory,where“obvious”notionssuchasintersectioncanbesomewhatsurprising.Theotheraspectisactuallyprovingpropertiesofα/β-groupsandtheirsubgroups.Whiledoingso,weshalltrytoskiptediousstep-by-stepveriﬁcationsinthisnewcontextDate:June14,2006.2000MathematicsSubjectClassiﬁcation.03C45,03C95.Keywordsandphrases.simpletheories,almosthyperdeﬁnablegroups.TheauthorwouldliketothankFrankO.Wagnerforhelpfuldiscussionsandcomments.Atthetimeofthewritingofthispaper,theauthorwasagraduatestudentwiththeE´quipedeLogiqueMathe´matiqueofUniversite´ParisVII.1

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