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A Non-Quasiconvex Subgroup of a Hyperbolic Group with An New York Journal of Mathematics New York J Math A Nonquasiconvex Subgroup of a Hyp erb olic Group with an Exotic Limit Set Ilya Kap ovich Abstract We construct an example of a torsion free freely indecomp osable nitely presented nonquasiconvex subgroup H of a word hyp erb olic group G suchthatthe limit set of H is not the limit set of a quasiconvex subgroup of G In particular this gives a counterexample to the conjecture of G Swarup that a nitely presented oneended subgroup of a word hyp erb olic group is quasiconvex if and only if it has nite index in its virtual normalizer Contents Intro duction Some Denitions and Notations The Pro ofs References Intro duction A subgroup H of a word hyp erb olic group G is quasiconvex or rationalinG if for any nite generating set A of G there is suchthatevery geo desic in the Cayley graph G AofG with b oth endp oints in H is contained in neighborhood of H The notion of a quasiconvex subgroup corresp onds roughly sp eaking to that of geometric niteness in the theory of classical hyp erb olic groups see Swa KS Pi Quasiconvex subgroups of word hyp erb olic groups are nitely presentable and word hyp erb olic and their nite intersections are again quasiconvex Non quasiconvex nitely generated subgroups of word hyp erb olic groups are quite rare and there are very few examples of them We know only three basic examples of this sort The rst is based on a remarkable construction of E Rips R which allows one given an arbitrary nitely presented group Q to construct a word hyp erb olic group G andatwogenerator subgroup H of G suchthatH is normal in G and the quotient is isomorphic to Q The second example is based on the existence of a closed hyp erb olic manifold b ering over a circle provided by results of W Thurston and T Jorgensen The third example is obtained using the result Received June Mathematics Subject Classication Primary F Secondary E Key words and phrases hyp erb olic group quasiconvex subgroup limit set This research is supp orted by an Alfred P Sloan Do ctoral Dissertation Fellowship c State UniversityofNewYork ISSN A Nonquasiconvex Subgroup of M Bestvina and M Feign BF who proved that if F is a nonab elian free group of nite rank and is an automorphism of F without periodic conjugacy classes then the HNNextension of F along is word hyp erb olic If G is a word hyp erb olic group then we denote the boundary of G see Gr GH and CDP by G For a subgroup H of G the limit set H ofH is the G set of all limits in G of sequences of elements of H In this note we construct an example of a nonquasiconvex nitely presented oneended subgroup H of a word hyp erb olic group G such that the limit set of H is exotic By exotic we mean that the limit set of H is not the limit set of a quasiconvex subgroup of G This result is of some interest since in the previously known examples nonquasiconvex subgroups were normal in the ambienthyp erb olic groups and thus see KS had the same limit sets Our subgroup H also provides a counterexample to the conjecture of G A Swarup Swa which stated that a nitely presented freely indecomp osable subgroup of a torsionfree word hyp erb olic group is quasiconvex if and only if it has nite index in its virtual normalizer this statementwas known to b e true for dimensional Kleinian groups The subgroup H constructed here coincides with its virtual normalizer Here by the virtual normalizer of a subgroup H of a group G we mean the subgroup VN H fg G j jH H gH g j jgH g H gH g j g G Some Denitions and Notations A geodesic in a metric space X d is an isometric emb edding l X where l and l is a segment of the real line We say that a metric space X A X disgeodesic if anytwopoints of X can b e joined by a geo desic path in path l X is called quasigeodesic if it is parametrized by its arclength and for any t t l jt t j d t t If x y and z are p oints in a metric space X dwe set x y dz xdz y dx y z The quantityx y is called the Gromov inner product of x and y with resp ect to z z Let b e a triangle in a metric space X d with geo desic sides and and vertices x y z See Figure We say that the points p q r on and are the vertices of the inscribed triangle for if dx pdx r y z dy pdy qx z and dz r x y dz qx y In this situation is called thin if for each t dx p z dp r where p r are points on with dx p dx r t and if the symmetric condition holds for y and z A geo desic metric space X discalled hyperbolic if there is such that all geo desic triangles are thin Ilya Kapovich y α p q β p’ r’ r γ z x Figure If G is a nitely generated group and G is a nite generating set for G we denote the Cayley graph of G with resp ect to G byG G and denote by d the G word metric on G G Also for any g G we dene the word length of g as l g d g It is easy to see that G G d is a geo desic metric space If w G G G is a word in the generators G we denote by w the elementofG which w represents A nitely generated group G is called word hyperbolic if for each nite generating set G for G there is suchthattheCayley graph G G with the word metric d is hyp erb olic G A subgroup H of a word hyp erb olic group G is called quasiconvex in G if for some and therefore for any nite generating set G of G there is such that every geo desic in the Cayley graph G G ofG with b oth endp oints in H lies in the neighb orho o d of H say that a If G is a word hyp erb olic group with a nite generating set G we sequence of p oints fg G j n N g denes a point at innity if n lim inf g g i j n ij n where the Gromov inner pro duct is taken in d metric Two sequences a G n nN and b dening p oints at innity are called equivalent if n nN lim inf a b i j n ij n The boundary G of G is dened to b e the set of equivalence classes of sequences G is the equivalence class of a sequence a dening p oints at innity If a n nN wesay that a converges to a and write lim a a The b oundary G can n nN n n be endowed with a natural top ology which makes it a compact and metrizable space It turns out that the denition of G and the top ology on it are indep endent of the choice of the word metric for G Moreover G acts on G by homeomorphisms and the action is given by g lim a lim ga where g G and a n n n n n nN denes a p oint at innity If S is a subset of G eg a subgroup of G we dene the limit set S to b e the set of limits in G of sequences of elements of S G A Nonquasiconvex Subgroup The Pro ofs Prop osition A Let F be the fundamental group of a closedhyperbolic surface S Let be an automorphism of F induced by a pseudoanosov homeomorphism of S Take G to be the mappingtorus group of thatis G hF t j tf t f f F i Let x F be an element which is not a proper power in F and so obviously x is not a proper power in G Let G be acopy of G The group G contains a copy F of F and a copy x of x Set M G G xx and H sg pF F Then M is torsionfree and wordhyperbolic H is nitely presented freely indecomposable and nonquasiconvex in M Pro of The group G is torsion free and word hyp erb olic since it is the fundamental group of a closed manifold of constant negativecurvature see Th Thus M is word hyp erb olic by the combination theorem for negatively curved groups see BF KM Notice that H sg pF F F F and so H is torsionfree x x nitely presentable and freely indecomp osable Moreover H is word hyp erb olic by the same combination theorem Supp ose H is quasiconvex in M It is shown in BGSS that F is rational with resp ect to some automatic structure on H since H F F where C hxi hx i C is cyclic Therefore F is quasiconvex in H see for example Swa Thus since H is quasiconvex in M and F is quasiconvex in H the subgroup F is quasiconvex in M However F is innite and has innite index in its normalizer in M which see KS implies that F is not quasiconvex in M This contradiction completes the pro of of Prop osition A Theorem B Let G G M andH beasinProposition A Let K be the limit set H of H in the boundary M of M Then M H Stab K ff M j fK K g M Before pro ceeding with the pro of wecho ose a nite generating set G for G and its copy G Then G denes the word length l and the word metric d for G G G Analogously GG is a nite generating set for M G G which denes the C w ord length l and the word metric d on M Fix a such that all d M M M geo desic triangles are thin We also denote by C the subgroup of M generated by x x The element x x will sometimes b e denoted by c Thus M G G C and H F F We need to accumulate some preliminary information b efore C pro ceeding with the pro of of Theorem B Ilya Kapovich h V Ui i Vj Uj ui vi uj vj y V U x v u z w hk 1 Figure Lemma Let A be a word hyperbolic group with a xed nite generating set S Let H and K be quasiconvex subgroups of A such that for every a A aH a K fg Then thereisaconstant r such that for every h H and k K l hk r l k A A Pro of Fix a nite generating set H for H and a nite generating set K for K suchthatevery geo desic h Since K and H are quasiconvex in A there is h H in the Cayley graph of A lies in the neighborhood of H and every geo desic k k K lies in the neighb orho o d of K Also there is suchthatevery d geo desic d geo desic word denes a quasigeo desic in the Cayley graph of H K A Let h H and k K Fix d geo desic representatives u v w of h k and G hk resp ectively Also x a d geo desic representative U of h and a d geo desic H K representative V of k Consider the geo desic triangle in the Cayley graph of A with sides u v
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