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Test Elements and the Retract Theorem in Hyperbolic Groups

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Test Elements and the Retract Theorem in Hyperbolic Groups � New York Journal of Mathematics New York J Math Test Elements and the Retract Theorem in Hyp erb olic Groups John C ONeill and Edward C Turner Abstract We prove that in many p erhaps all torsion free hyp erb olic groups test elements are precisely those elements not contained in prop er retracts We also show that all Fuchsian groups have this prop erty Finallyweshow that all surface groups except Z Z have test elements Contents Intro duction Torsion free hyp erb olic groups The Retract Theorem in Fuchsian groups Test elements in surface groups References Intro duction Denition If G is a group then g G is a test element if for any endomorphism G G g g implies that is an automorphism The notion of test elements was rst considered in the context of free groups in which they are called test words The rst example was provided in by Nielsen N who showed that the basic commutator a b aba b is a test word in the free group F a b Considerable progress has b een made recently in understanding b oth test elements and test wordssee Tu for example and the references cited there The following reformulation of the denition makes clear how test elements are used to recognize automorphisms g is a test elementif g g for some automorphism implies that is an automorphism Thus the issue of deciding whether is an automorphism is replaced by that of deciding whether g andg are equivalent under the action of the automorphism group AutG The classic algorithm of J H C Whitehead W decides very eectively when two elements of a free group F are equivalent under the action of AutF in a forthcoming Received February Mathematics Subject Classication E F Key words and phrases test words test elements endomorphisms automorphisms hyp erb olic groups ISSN John C ONeil l and Edward C Turner pap er D J Collins and the second author will showhow to do the same thing in a torsionfree hyp erb olic group Pro ducing nontest elements is quite easy Supp ose for example that the sub group R of G is a prop er retract ie the image of a nonsurjectivemap G G called a retraction with the prop ertythat r r for all r R Then no element of R can b e a test element In Tu the following theoremthe Retract Theorem for freegroupswas proven characterizing test words in free groups Theorem A word w in a free group F is a test word if and only if w is not in any proper retract We are interested in deciding whether the Retract Theorem is true for gen eral hyp erb olic groups By hyp erb olic we mean word hyp erb olic in the sense of Gromovsee eg GH In particular hyp erb olic groups are nitely generated We succeed in proving it for torsion free hyp erb olic groups that are stably hyperbolic in the following sense Denition A hyp erb olic group is stably hyperbolic if for every endomorphism n G G there are arbitrarily large values of n so that Gishyp erb olic Anyhyp erb olic group with the prop ertythat every nitely generated subgroup is hyp erb olic hyp erb olic surface groups for example clearly satises this prop erty A famous application of the Rips construction R shows that some hyp erb olic groups subgroups We mo dify this construction have nitely generated nonhyp erb olic in Section to pro duce an example in which such a subgroup is the image of an endomorphism Its relatively easy to extend this to nd endomorphisms of hyp erb olic groups that have arbitrarily many nonhyp erb olic forward images but in all cases weve studied the images are eventually hyp erb olic It seems quite p ossible that all hyp erb olic groups are actually stably hyp erb olic Our sp ecic results are the following Theorem If G is a torsion free stably hyperbolic group and g G then g is a test element if and only if g is not in any proper retract uchsian group and h H then h is a Theorem If H is a nitely generated F test element if and only if h is not in any proper retract This applies in particular if H is a nite freeproduct of cyclic groups Theorem If G is a surfacegroup other than Z Z then G has test elements The situation for surface groups is interestingall surface groups except Z Z have test elements explicit examples given in Section and all except the funda mental group of the Klein b ottle ha b j aba b i satisfy the Retract Theorem In V it was shown that b lies in no prop er retract but is nevertheless not a test word since it is xed by aa bb We will use the following terminology Denition If G is a group and G G is an endomorphism then n n n j G G and n G j G G G T n where G i The group G is Hopan if it is not isomorphic to any of its prop er quotients and is coHopan if it is not isomorphic to any of its prop er subgroups Test Elements and the Retract Theorem in Hyperbolic Groups Torsion free hyp erb olic groups This section is devoted to a pro of of Theorem Our pro of of Theorem is mo deled on the pro of of the Retract Theorem in Tu the main technical to ol b eing Prop osition whose pro of we defer to the end of the section Prop osition Suppose that G G G G is a freeproduct of innite m cyclic and freely indecomposable coHopan groups If G G is a monomor phism then G is a freeoffactor G Prop osition If G is a torsionfree hyperbolic group and G G is an n endomorphism with the property that G is hyperbolic for arbitrarily large n N then G is a free factor of G for some N Pro of It was proven by Sela Se that if G G is an endomorphism of a n n n torsionfree hyp erb olic group G then j G G is a monomorphism G n for large enough n Let n b e large enough so that j is a monomorphism and G n n that G is hyp erb olic and consider the free pro duct decomp osition G H H into freely indecomp osable hyp erb olic factors Sela Se has also m proven that every freely indecomp osable torsion free hyp erb olic group is either coHopan or innite cyclic Prop osition now follows from Prop osition Pro of of Theorem The fact that elements of prop er retracts are not test el ements is trivially true in all groups since a prop er retraction is not an automor phism To prove the converse we b egin with the following general observation if G G is a monomorphism of a group G then is an automorphism of G It is clear that is injective To see that is surjective let g G n then for every n there exists g Gsuchthat g g since is injective n n g g for all m n Hence g G and is surjective m n Now supp ose that G is a stably hyp erb olic group that g is not a test element and that is a endomorphism which is not an automorphism so that g g we show that g lies in a prop er retract byshowing that G is a prop er retract n For large enough n is a monomorphism by Se since is an automorphism by the observation ab ove According to Prop osition Gisa N N free factor of G for some N cho ose suchanN and let G G b e a free factor pro jection mapping Then N N G G is a retraction mapping Thus is a retract It may b e that the Retract Theorem is true for general hyp erb olic groups Theo rem is a partial result in this direction It may also b e the case that all hyp erb olic groups are stably hyp erb olic However the following example suggested byGA Swarup shows that it may b e that Gisnothyp erb olic but in this case G is trivial Example Supp ose that F F h x x x x j x x x x x x x x i and let i F F F F by x x for i John C ONeil l and Edward C Turner In Dr it was shown that ker has rank and is not nitely presented and is therefore not a hyp erb olic group Nowlet G F F F F hy y i and G F hx y y i r r r r r by x x i i y y i i ker Then P has rank and let P Now the Rips construction R pro duces a hyp erb olic group H with rank r that maps onto G by a map with kernel generated bytwo added generators a and b Then H P isanonhyp erb olic subgroup of H of rank at most P H ha bi H G F r Nowlet r F H b e a surjection and H H Then H r is hyp erb olic and is an endomorphism whose image H is not hyp erb olic Pro of of Prop osition We begin by observing that it suces to show that n G is a free factor of G for some n in fact we will show that this is n true for all suciently large n For if G G G for some G then the monomorphism n n G G pulls this factorization backto G n n G G G n But it is straightforward to show that G G The pro of is a geometric generalization of the Takahasi Theorem Ta and is mo deled on the pro of of the Takahasi result outlined in problem on page of MKS We will need the following slightvariant of the usual normal form measure of length in a free pro duct which dep ends on the pro duct decomp osition Denition If G G G G is a free pro duct of freely indecomp osable m factors then the length jg j is G t X jg j jg j j G G i j j i form g g g g g G for some i where g has free pro duct normal t j i j j th jg j Z and jg j if G n if G Z and g is the n power of a j G j G i i j i i j j j j generator Test Elements and the Retract Theorem in Hyperbolic Groups We order the factors of G so that G Z for i thus i G F G G m Let X b e the natural complex with fundamental group G whichisthewedge of circles and complexes K K K G each of whichisjoinedto m i i the wedge p ointby a line segment The nested sequence of subgroups n G G G G G determines a sequence of coverings of X p p p p k k k X X X X X k k k where X G Let P p p p X X k k k k In the covering
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