Deformations of the discrete Heisenberg group

Transcription

1 No. 4] Proc. Japan Acad., 89, Ser. A (3) 55 Deformations of the discrete Heisenberg group y Severin ARMEIER Instituto de Matemática, Estatstica e omputação ientfica, Universidade Estadual de ampinas, Rua Sérgio uarque de Holanda, 65, idade Universitária Zeferino Vaz, ampinas, SP, razil (ommunicated by Masaki KASHIWARA, M.J.A., March, 3) Abstract: We study deformations of the discrete Heisenberg group acting properly discontinuously on the Heisenberg group from the left and right and obtain a complete description of the deformation space. Key words: Deformation; discrete group; properly discontinuous action; homogeneous space; Heisenberg group.. Introduction and statement of main result. We will be interested in deformations of the discrete Heisenberg group as a group acting properly discontinuously and cocompactly on a space X. The following defines our notion of deformation. Definition. ([K93, K, KN6]). Let G be a Lie group acting continuously on a locally compact space X and let G be a discrete subgroup. Define the parameter space of deformations of within G, acting properly discontinuously on the space X as 8 9 is injective; >< ð Þ acts properly >= Rð ;G; XÞ ¼ :! G discontinuously >: >; and freely on X and the deformation space as T ð ;G; XÞ ¼Rð ;G; XÞ=G; where G acts on Rð ;G; XÞ by conjugation, so that T ð ;G; XÞ is the space of non-trivial deformations. There is a natural topology on the parameter space Rð ;G; XÞ as a subset of Homð ;GÞ endowed with the compact open topology. We then consider the quotient topology on the deformation space T ð ;G; XÞ ([K93, K]). If X is an irreducible Riemannian symmetric space G=K, Selberg Weil rigidity ([W64]) states that T ¼ T ð ;G; G=KÞ is discrete if and only if G is not locally isomorphic to SL R.Anexampleof the failure of rigidity is when G ¼ PSL R, is the Mathematics Subject lassification. Primary 3G8, E4, F3, 57S3. fundamental group of a Riemann surface of genus g > and X ¼ SL R=SO is the Poincaré disk. Then T is the Teichmüller space, which has dimension 6g 6. The study of deformations of discontinuous groups for non-riemannian homogeneous spaces and the failure of rigidity was initiated by Kobayashi [K93]; Kobayashi [K98] treats the case when G is semi-simple. A complete description of the parameter and deformation spaces was first given for ¼ Z k acting on X ¼ R kþ via some nilpotent group of transformations G in [KN6] and these results were extended to the case where G is the Heisenberg group, H is any connected Lie subgroup and is a subgroup acting properly discontinuously and freely on X ¼ G=H, in [KY]. In this paper, we give a concrete description of the space Rð ;G G; GÞ,whereGis the Heisenberg group, ¼ G \ GL 3 Z is the discrete Heisenberg group and the direct product group G G acts on the group manifold G from the left and right. Our main result is the following. Theorem.. For the deformation space T ð ;G G; GÞ of the discrete Heisenberg group acting properly discontinuously on the group manifold G from the left and right, we have the homeomorphism T ð ;G G; GÞ ¼ GL R R R 3 :. Notation. Let G denote the Heisenberg group and ¼ G \ GL 3 Z denote the discrete Heisenberg group. We will replace the matrix notation by defining doi:.379/pjaa #3 The Japan Academy

2 56 S. ARMEIER [Vol. 89(A), 3 a a c 4 b 5 b A: c We will fix a presentation ¼h ; i,where 3 3 ðþ ¼ 4 5 and ¼ 4 5: As a subgroup always acts properly discontinuously and freely on G from the left and the quotient space ng is a compact manifold. Similarly always acts properly discontinuously from the right with compact quotient G=. To let act both from the left and from the right, we rewrite G as the homogeneous space G G=G, where : G! G G is the diagonal embedding. Then acts on G G=G via homomorphisms! G G. We note here that Homð ;G GÞ ¼ ðg GÞðGGÞ as sets, because each generator ; can be assigned any element in G G, asanyrelations and satisfy as elements of G are also satisfied by any two arbitrary elements in G G. Via the topology on G, then, Homð ;G GÞ can be regarded a topological space. In particular, for G being the Heisenberg groupwehavethatg¼ R 3, whence Homð ;G GÞ ¼ R. Any homomorphism! G G can be written as a pair of homomorphisms ; :! G. Nowwrite ; ¼fððÞ; ðþþ j g for the image of the pair ð; Þ:! G G. Then acts on G G=G via ; and the action of on G as subgroup (on the left) is recovered as the action of id; on G G=G, where id is the inclusion and is the trivial homomorphism. However, for general ; this action is not necessarily properly discontinuous. Remark.. Rewriting G as G G=G for G ¼ SL g R allowed Goldman [G85] to construct non-standard Lorentz space forms. Goldman s conjecture concerning the existence of an open neighbourhood of the embedding id, throughout which the group action remains properly discontinuous was resolved affirmatively for reductive Lie groups by Kobayashi [K98]. An analogous result holds if G is a simply connected Lie group and is a cocompact discrete group by an unpublished result of T. Yoshino. Our results below show this feature explicitly for G being the Heisenberg group. 3. Property (I) and proper actions. To check for proper discontinuity of the action of ;, we will use a criterion by Nasrin [N] for -step nilpotent groups, which relates properness to the property (I). Definition 3. ([K9], Def. 6). We say the triplet ðl; H; GÞ has the property (I) if L \ ghg is compact for any g G. (See [L95] for the relationship between the property (I) and proper actions in the more general context of locally compact topological groups acting on locally compact topological spaces.) Theorem 3. ([N], Thm..). Let G be a simply connected -step nilpotent Lie group, and let H and L be connected subgroups. Then the following conditions are equivalent. (a) L acts properly on G=H, (b) the triplet ðl; H; GÞ has the property (I), (c) L \ ghg ¼feg for any g G. We will apply this theorem to the triple ðl ; ; G; G GÞ, whereg is again the Heisenberg group and L ; is the extension of ; defined as follows. Definition 3.3. Let be a discrete subgroup in a Lie group G. A connected subgroup L G is said to be the extension of if L contains cocompactly. The following lemma will allow us to use Thm. 3. to determine the conditions under which ; acts properly discontinuously. Lemma 3.4 ([K89]). Let L be a Lie group acting continuously on a locally compact space X. Let L be a discrete subgroup such that nl is compact. Then the following conditions are equivalent. (a) acts properly discontinuously on X, (b) L acts properly on X. 4. Main results. To find the extension of ;, we use the (global) diffeomorphism exp: g! G, whose inverse we denote by log. Let; :! G be any two homomorphisms. Then and are determined by their values on the generators, which (in the notation of ) we will set to be 3 a i a 3 i ð i Þ¼4 5 and ðþ ð i Þ¼4 5; b i c i for i ¼ ;. Now, let : g! g be a Lie algebra homomorphism defined on the generators by ðlog i Þ¼log ð i Þ, for i ¼ ;, and ð½log ; log ŠÞ ¼ log ð½ ; ŠÞ, and extended linearly; let b i c i

3 No. 4] Deformations of the discrete Heisenberg group 57 : G! G be defined by ¼ exp log. Then j ¼, sothatextends in the sense that is defined on all of G. Ifwewrite for the extension of to all of G, then L ; ¼fððgÞ; ðgþþ j g Gg is the extension of ; in the sense of Def Next, we will check condition (c) of Thm. 3. for ðl ; ; G; G GÞ. Wehavethat L ; \ðg ;g ÞGðg ;g Þ ¼feg ð3þ, ðgþ ¼g g ðgþðg g Þ only if g ¼ e, ðlog gþ ¼Ad g g ðlog gþ only if log g ¼ for all ðg ;g ÞGG. Nowwrite a c a c ab g b A and log g b A: alculating the LHS and RHS of (3) explicitly, it follows that (3) is equivalent to a a b b A@ b A a b a b c a a a ¼ b A@ b A ) a ¼ b ¼ c ¼ : a b a b c Writing A ¼ a a and A ¼ a a ð4þ b b b b ; we can rewrite condition (3) as A A det det A det A 6¼ ; and we obtain the following proposition. Proposition 4.. The group ; acts properly discontinuously and cocompactly on G G=G if and only if the following two conditions hold. (a) detða A Þ 6¼, and (b) det A det A 6¼, where A; A are determined by ; via () and (4). Proper discontinuity is contained in the above argument. For cocompactness we make use of the following lemma. Lemma 4.. Let be as in () and A be defined by (4). Thendet A 6¼, is injective. Proof. det A is precisely the ð; 3Þ entry of the commutator ½ð Þ;ð ÞŠ and det A 6¼ if and only if the image ð Þ is non-commutative. We show that ð Þ being non-commutative is equivalent to being injective. If is injective, ð Þ ¼ is non-commutative. onversely, write N ¼ ker and assume that ð Þ is non-commutative. We have the commutative diagram N Z N N/N Z Z Γ Z Z/N Z Γ/N Γ/ZN whose rows and columns are exact by the nine lemma. Turning our attention to the first column, the top left entry N \ Z can be considered as a subgroup of Z, and is thus equal to (i),(ii) Z, or (iii) mz, forsomem. ase (ii). If N \ Z ¼ Z, N contains the commutator Z ¼½ ; Š, contradicting the fact that ð Þ ¼ =N was assumed non-commutative. ase (iii). If N \ Z ¼ mz, for m, then Z=N \ Z ¼ Z m in the bottom left entry. However, Z m is finite and contains torsion elements and injects into =N. y the first isomorphism theorem for groups, the induced map : =N! G is injective. ut G is torsion-free, whence =N is torsion-free also and we obtain a contradiction. We conclude that N \ Z ¼ (case (i)). Now, write :! Z for the projection and : N! N=N \ Z for the restriction of to N. Let be any element in and n N. SinceN is normal, n N. Then ðn Þ¼ ðþ ðnþ ð Þ¼ ðnþ; where the last equality follows from the fact that im injects into Z and is therefore commutative. Since N \ Z ¼, is an isomorphism and we conclude that n ¼ n, i.e.n is contained in the centraliser Z. Then N \ Z ¼ shows that N is trivial, whence is injective.

4 58 S. ARMEIER [Vol. 89(A), ProofofProp.4..Using Thm. 3., we have shown that L ; acts properly on G G=G if and only if conditions (a) and (b) hold. Applying Lem. 3.4, L ; acts properly on G G=G if and only if ; acts properly discontinuously on G G=G. y Lem. 4., condition (b) shows that at least one of ; must be injective, whence the cohomological dimension cd ; ¼ 3. It is a fact, based on a standard argument invoking Poincaré duality, that if a group acts (faithfully) on a contractible manifold X and cd ¼ dim X, then nx is compact (cf. [K89], or. 5.5). Since G G=G ¼ R 3 is indeed contractible and dim G G=G ¼ cd ; ¼ 3, the double quotient ; ng G=G is compact. Prop. 4. can be turned into a method for determining pairs of homomorphisms for which ; acts properly discontinuously and cocompactly on G fromtheleftandrightasfollows. Let S ¼ s s GL R s s 3 and ðs; t ;t ;t ;t 3 ;c ;c ;c ;c ÞGL R R R 7. Define a map ð5þ :GL R R R 7! Rð ;G G; GÞ ðs;t ;t ;t ;t 3 ;c ;c ;c ;c Þ 7! ; where ¼ð; Þ is defined by ðs 3 ðt þ t 3 Þþs þ s t Þ 6 ð Þ¼ ðs 7 4 ðt þ t 3 Þþs þ s 3 t Þ 5; c ðs 3 ðt t 3 Þþs þ s t Þ 6 ð Þ¼ ðs 7 4 3ðt t 3 Þþs 3 þ s t Þ 5; c ðs 3 ðt þ t 3 Þ s þ s t Þ 6 ð Þ¼ ðs 7 4 ðt þ t 3 Þ s þ s 3 t Þ 5; c ðs 3 ðt t 3 Þ s þ s t Þ 6 ð Þ¼ ðs 7 4 3ðt t 3 Þ s 3 þ s t Þ 5: c Determining A; A via (4), one checks that A A ¼ S and det A det A ¼ t det S 6¼, as t R. Thus, conditions (a) and (b) from Prop. 4. are satisfied and ; acts properly discontinuously and cocompactly on G from the left and right. Moreover,wehavethefollowingtheorem. Theorem 4.3. The map (see (5)) induces a homeomorphism from GL R R R 7 onto the parameter space Rð ;G G; GÞ of deformations of acting properly discontinuously on the group manifold G from the left and right. Furthermore, the deformation space T ð ;G G; GÞ is homeomorphic to GL R R R 3. Proof. The idea of the proof and the origin of the map is the following. The space of pairs of matrices satisfying (a) and(b)ofprop.4.canbedeterminedasfollows. Suppose A; A satisfy (a) and (b). onsider the map!: ða; A Þ 7! ðu;vþ¼ða A ; ða A Þ ða þ A ÞÞ; which is well-defined, since U ¼ A A is invertible. We can find an inverse mapping : ðu;vþ 7! ð ðuv þ UÞ; ðuv UÞÞ and one checks that! ¼ id and! ¼ id. For U and V, condition (a) is equivalent to the condition that U GL R; condition (b) translates into the condition det ðuv þ UÞ 6¼ det ðuv UÞ, detðv þ IÞ 6¼ detðv IÞ;, det V þ tr V 6¼ det V tr V, tr V 6¼ ; where I denotes the identity matrix. Then, writing M ¼fV M ðrþ jtr V 6¼ g ¼ R R 3, the map :GL R M!fðA; A ÞjA; A satify (a) & (b)g is a homeomorphism. Writing id for the identity on R 4 ¼fðc ;c ;c ;c Þjc ;c ;c ;c Rg, id ¼ is the homeomorphism :GL R R R 7! Rð ;G G; GÞ up to the identification M ¼ R R 3. The conjugation action of G G on ; leaves the superdiagonal entries of each factor unchanged and is transitive on the ð; 3Þ entries, so that T ð ;G G; GÞ ¼Rð ;G G; GÞ=ðG GÞ is homeomorphic to GL R R R 3 : 5. Geometric interpretation of main result. Geometrically speaking, we have the central extensions

5 No. 4] Deformations of the discrete Heisenberg group 59! R! G! R!! Z!! Z! and by quotienting ng canbeviewedasacircle bundle over the torus. The two conditions of Prop. 4. can then be interpreted as follows. The matrix A A determines a Riemannian structure on this torus and det A det A determines the structure on (i.e. length of) the circle. In particular, the number of connected components (which equals four) of the deformation space T ð ;G G; GÞ corresponds to the number of possible combinations of orientations on the torus and the circle. Example 5.. Let 3 3 ð Þ¼4c5; ð Þ¼45 and 3 3 ð Þ¼4c5; ð Þ¼45: Letting c vary from to, we obtain a family of groups ; (which lies in the component of both base space and fibre orientations being positive), which by Prop. 4. act cocompactly and properly discontinuously on G G=G, wherethelengthof the fibre varies from 3 to and the structure on the torus remains unchanged and is given by the matrix ð Þ. Similarly, it is possible to find families of groups, which only change the structure on the base space, leaving the length of the fibre unchanged; or families, for which both the structure on the base space and the length of the fibre are fixed, but the connection form is deformed. Remark 5.. General examples, like the one above, stand in contrast to the case when G is semisimple of real rank e.g. G ¼ SL R, SOðn; Þ, SUðn; Þ, Spðn; Þ for which any properly discontinuous group for G G=G is a graph up to a finite-index subgroup ([K93], Thm. and Rmk. ). Acknowledgements. The author would like to thank Prof. Taro Yoshino for detailed comments on an earlier version of this paper and Prof. Toshiyuki Kobayashi for his comments, guidance and invaluable advice. References [ KY ] A. aklouti, I. Kédim and T. Yoshino, On the deformation space of lifford Klein forms of Heisenberg groups, Int. Math. Res. Not. IMRN 8, no.6,art.idrnn66,35pp. [ G85 ] W. M. Goldman, Nonstandard Lorentz space forms, J. Differential Geom. (985), no., [ K89 ] T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 85 (989), no., [ K9 ] T. Kobayashi, Discontinuous groups acting on homogeneous spaces of reductive type, in Representation theory of Lie groups and Lie algebras (Fuji-Kawaguchiko, 99), 59 75, World Sci. Publ., River Edge, NJ, 99. [ K93 ] T. Kobayashi, On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geom. Phys. (993), no., [ K98 ] T. Kobayashi, Deformation of compact lifford Klein forms of indefinite-riemannian homogeneous manifolds, Math. Ann. 3 (998), no. 3, [ K ] T. Kobayashi, Discontinuous groups for non- Riemannian homogeneous spaces, in Mathematics unlimited and beyond, , Springer, erlin,. [KN6] T. Kobayashi and S. Nasrin, Deformation of properly discontinuous actions of Z k on R kþ, Internat.J.Math.7 (6), no., [ L95 ] R.L.Lipsman,Properactionsandacompactness condition, J. Lie Theory 5 (995), no., [ N ] S. Nasrin, riterion of proper actions for -step nilpotent Lie groups, Tokyo J. Math. 4 (), no., [ W64 ] A. Weil, Remarks on the cohomology of groups, Ann. of Math. () 8 (964),