Presentations for cusped arithmetic hyperbolic lattices

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1 Presentatons for cused arthmetc hyerbolc lattces Alce Mar, Julen Pauert October 5, 7 Abstract We resent a general method to comute a resentaton for any cused hyerbolc lattce Γ, alyng a classcal result of Macbeath to a sutable Γ-nvarant horoball cover of the corresondng symmetrc sace. As alcatons we comute resentatons for the Pcard modular grous PU(,, O d ) for d =, 3, 7 and the quaternonc lattce PU(,, H) wth entres n the Hurwtz nteger rng H. Introducton Dscrete subgrous and lattces n semsmle Le grous form a rch and well-studed class of fntely generated grous actng on non-ostvely curved metrc saces. The case of real ran one, where the assocated symmetrc sace s negatvely curved, s of secal nterest. There are essentally two man famles of constructons of such lattces, arthmetc on one hand and geometrc on the other. Arthmetc lattces are roughly seang obtaned by tang matrces wth entres lyng n the nteger rng of some number feld; the general defnton s more comlcated and we wll not gve t here, as the arthmetc lattces that we consder n ths aer are of ths smlest tye. By Marguls celebrated suerrgdty and arthmetcty theorems, all (rreducble) lattces n G are of ths arthmetc tye when G s a semsmle Le grou of real ran at least. The other famly nvolves geometrc constructons such as olyhedra, reflectons or other tyes of nvolutons or other fnte-order sometres. A rototye of ths tye of constructon s gven by Coxeter grous n the constant curvature geometres E n, S n and H n, whch are generated by reflectons across hyerlanes. These grous are classcal and were classfed by Coxeter n the saces E n and S n, whereas ther hyerbolc counterarts (studed by Vnberg and others) are stll not comletely understood. However by constructon these grous come equed wth data ncludng a resentaton (as an abtract Coxeter grou) and a fundamental doman for ther acton on the symmetrc sace. Arthmetc lattces are gven by a global descrton and ther global structure s n some sense well understood by wor of Segel, Borel, Tts, Prasad and others. However concrete nformaton such as a resentaton and a fundamental doman are not readly accessble from the arthmetc constructon. One can obtan geometrc nformaton such as volume by Prasad s celebrated volume formula ([P]) but comutng the constants aearng n ths formula usually nvolves some non-trval wor (see for examle [Be] and [Sto]). Very few resentatons of arthmetc lattces, and of lattces n general, are nown. Presentatons can rovde useful geometrc and algebrac nformaton about grous, such as exlct ndex of torson-free subgrous (effectve Selberg lemma, as used for examle n [Sto]), cohomology of the grou Γ or quotent sace X/Γ, see for nstance [Y] (for the Pcard modular grous wth d =, 3) and of course reresentatons of Γ, for nstance f one s nterested n deformatons of Γ nto a larger Le grou. Presentatons for SL(n, Z) wth n 3 were gven by Stenberg ([Ste], followng Magnus); the case of SL(, Z) s classcal and ossbly dates to Gauss; see also Segel [S]. In ran one, Swan gave n [Sw] resentatons for the Banch grous PGL(, O d ) (where O d denotes the rng of ntegers of Q[ d] for d a ostve square-free nteger), followng Banch s orgnal constructon n [B]. These act as sometres of (real) hyerbolc 3-sace, as they are lattces n PGL(, C) Isom + (H 3 R ). Presentatons for the related Pcard modular grous PU(,, O d ) were found only recently n the smlest cases of d = 3 ([FP]) and d = ([FFP]). One of the reasons for ths s that the assocated symmetrc sace, comlex hyerbolc -sace H C, s more comlcated and n artcular has non-constant (nched) negatve curvature. A artcular feature of such saces, the absence of totally geodesc real hyersurfaces, maes constructons of fundamental domans dffcult as there are no obvous walls to use to bound such domans. The

2 resentatons obtaned for d =, 3 were n fact obtaned by constructng fundamental domans and usng the Poncaré olyhedron theorem. Ths aroach seems to become too comlcated when consderng more comlcated grous, such as Pcard modular grous wth hgher values of d, and no further such constructons have aeared. Usng a smlar strategy, Zhao gave n [Zh] generatng sets for the Pcard modular grous wth d =,, 3, 7, but he does not go as far as obtanng a resentaton, fndng a set whose translates covers the sace but wthout control over ntersectons and cycles. (We wll n fact use a coverng argument closely related to the one he uses to cover a fundamental rsm on the deal boundary by sometrc sheres, see Lemma 3). In ths aer we resent a method for obtanng resentatons for cused hyerbolc lattces,.e. noncocomact lattces n semsmle Le grous of real ran one, based on a classcal result of Macbeath (Theorem below) whch gves a resentaton for a grou Γ actng by homeomorhsms on a toologcal sace X, gven an oen subset V whose Γ-translates cover X. We aly ths by fndng a sutable horoball V based at a cus ont of Γ whose Γ-translates cover X, then analyzng the trle ntersectons and assocated cyles to obtan a resentaton for Γ. The man tools for ths analyss come from the addtonal arthmetc structure that we get by assumng that Γ s n fact an ntegral lattce n the sense that t s contaned n GL(n +, O E ) for some number feld E (or fntely generated dvson algebra over Q). The crucal such tool that we use s the noton of level between two E-ratonal boundary onts n X (see Defnton ) whch gves a noton of dstance between such onts usng only algebrac data. More mortantly for us, levels measure the relatve szes of horosheres based at the corresondng boundary onts, whch allows us to control whether or not such horosheres ntersect at a gven heght (see Lemma 4). As alcatons of the method we comute resentatons for the Pcard modular grous PU(,, O d ) wth d = 3,, 7, gven n (), (3) and (4). We exect to treat the cases d =, wth the same method but they are comutatonally more ntensve and wll be treated elsewhere. We also comute a resentaton (5) for the quaternon hyerbolc lattce whch we call the Hurwtz modular grou PU(,, H), where H s the rng of Hurwtz ntegers H = Z[,,, +++ ] H. Ths s a lattce n PU(,, H) (sometmes denoted PS(, )), actng on the 8-dmensonal symmetrc sace H H. As far as we now ths s the frst resentaton ever found for a hgher-dmensonal quaternon hyerbolc sace. (In dmenson, H H H4 R and such grous have been studed e.g. n [DVV], see also [A], [W] and [Ph]). The aer s organzed as follows. In Secton we dscuss generaltes about horoball coverngs of hyerbolc saces, levels for cus onts of ntegral lattces, and outlne how we aly Macbeath s theorem n ths context. In Secton 3 we dscuss horoshere ntersectons n more detal, n artcular the quanttatve relaton between levels and heghts of horosheres for ntegral lattces. In Sectons 4 and 5 we aly ths method to comute resentatons for the Pcard and Hurwtz modular grous resectvely. We would le to than Danel Allcoc for suggestng ths method and for many helful comments, and Matthew Stover and Danel Allcoc for ontng out a mstae n an earler verson of the aer. Horoball coverngs and lattce resentatons. Adated horoball coverngs and coverng comlex Let X be a negatvely curved symmetrc sace,.e. a hyerbolc sace H n K, wth K = R, C, H or O (and n f K = R, n = f K = O). We refer the reader to [CG] for general roertes of these saces and ther sometry grous. In artcular sometres of such saces are roughly classfed nto the followng 3 tyes: elltc (havng a fxed ont n X), arabolc (havng no fxed ont n X and exactly one on X) or loxodromc (havng no fxed ont n X and exactly two on X). Let Γ be a lattce n Isom(X); the well-nown Godement comactness crteron states that Γ contans arabolc sometres f and only f t s non-cocomact, whch we now assume. A cus ont of Γ s a ont of X fxed by a arabolc element of Γ; a cus grou of Γ s a subgrou of the form Stab Γ () where X s a cus ont of Γ. Assume that we are gven a Γ-nvarant coverng of X by (oen) horoballs (see Defnton 3),.e. a collecton B of horoballs such that: { γb B for all γ Γ and B B B B B = X

3 We wll moreover assume that each horoball B B s based at a cus ont of Γ, and that each cus ont of Γ s the baseont of a unque horoball n B (gvng a becton between cus onts of Γ and horoballs n B); we wll call such a coverng B a Γ-adated horoball coverng. Snce the lattce Γ has only fntely many cus onts modulo the acton of Γ, t follows that such a horoball coverng s a fnte unon of Γ-orbts of horoballs. Gven a Γ-adated horoball coverng B, the coverng comlex C(B) assocated to B s the smlcal -comlex wth vertex set B, wth an edge connectng each ar of vertces B and B such that B B, and a trangle for each trle of vertces B, B, B 3 such that B B B 3. Ths the -seleton of a smlcal comlex sometmes called the nerve of the coverng. By the above remar the quotent of the coverng comlex by the acton of Γ s a fnte smlcal -comlex. We wll use the followng classcal result of Macbeath ([M]): Theorem ([M]) Let Γ be a grou actng by homeomorhsms on a toologcal sace X. Let V be an oen subset of X whose Γ-translates cover X. () If X s connected then the set E(V ) = {γ Γ V γv } generates Γ. () If moreover X s smly-connected and V ath-connected, then Γ admts a resentaton wth generatng set E(V ) and relatons γ γ = γγ for all γ, γ E(V ) such that V γv γγ V. Now f as above Γ s a lattce n Isom(X) and B s a Γ-adated horoball coverng of X, we may as remared above wrte B as a fnte unon of Γ-orbts of horoballs B,..., B (say, mnmally). One can then aly Macbeath s theorem wth V = B... B, after ossbly enlargng each horoball B n order for ths unon to be (ath-)connected. For smlcty of exoston, we henceforth asume that Γ has a sngle cus, so that the Γ-adated horoball coverng conssts of a sngle Γ-orbt of horoballs (ths s the case n all examles consdered n ths aer). In that case the rocess of obtanng a resentaton fom the coverng comlex s closely related to a comlex of grous structure on the quotent of the coverng comlex, the only dfference beng that we need to tae nto account non-trval edge and face stablzers.. Levels and roxmal cus comlex Recall that f X s a hyerbolc sace H n K (wth K = R, C, H or O) then X admts the followng roectve model whch we brefly recall. Consder K n,, the vector sace K n+ endowed wth a Hermtan form, of sgnature (n, ). (When K = H or O we wll use the conventon that scalars act on vectors on the rght, whereas matrces act on vectors on the left.) Let V = { Z K n, Z, Z < }, V = { Z K n, Z, Z = } and let π : K n+ {} KP n denote roectvzaton. One then defnes H n K to be π(v ) KP n, endowed wth the dstance d (Bergman metrc) gven by, for Z, W V : cosh ( d(π(z), π(w )) ) Z, W = Z, Z W, W. () Note that the rght-hand sde s ndeendent of the choce of lfts Z, W. Then Isom (X) = PU(n,, K), the (roectvzaton of) the matrx grou reservng the Hermtan form (see [CG]). Note that PU(n,, K) s usually denoted PO(n, ) when K = R, PU(n, ) when K = C and PS(n, ) when K = H. The boundary at nfnty X s then dentfed wth π(v ) KP n. We would le to measure dstances between onts of X usng the Hermtan form as n (); one way to do ths s to use ntegral lfts of vectors wth ratonal coordnates as follows. We now assume that Γ s an ntegral lattce n the sense that t s contaned n U(H, O E ) for some number feld E (or fnte-degree dvson algebra over Q when K = H or O) wth rng of ntegers O E, and Hermtan form H =, defned over E. We say that an ntegral vector P = (,..., n+ ) O n+ E s rmtve f t has no ntegral submultle n the followng sense: f P λ O n+ E for some λ O E then λ s a unt n O E. If s an E-ratonal ont n KP n,.e. the roectve mage of a vector P = (,..., n+ ) E n+, a rmtve ntegral lft of s any lft P of to O n+ E whch s a rmtve ntegral vector. Lemma If O E s a rncal deal doman then rmtve ntegral lfts are unque u to multlcaton by a unt. 3

4 Lemma (a) Any column-vector of a matrx A U(H, O E ) s a rmtve ntegral vector. (b) If moreover E s magnary quadratc wth O E a rncal deal doman and one of the standard bass vectors B s H-sotroc, then for any H-sotroc rmtve ntegral vector V and A U(H, O E ), AV s a rmtve ntegral vector. Proof. (a) Let A U(H, O E ) and V a column-vector of A. Then V s ntegral; assumng that t s not rmtve, there would exst a non-unt λ O E such that V λ s also ntegral. But then the matrx A obtaned from A by relacng the column-vector V by V λ would also be n GL(n +, O E ), wth det A = det Aλ, a contradcton snce the latter s not an nteger, as det A s a unt and λ s not. (b) Let V be an H-sotroc rmtve ntegral vector and A U(H, O E ). If O E s a rncal deal doman then PU(H, O E ) has a sngle cus (see [Z]), therefore there exsts M U(H, O E ) mang B to AV λ for some λ E. Then as n (a) λ must be a unt, hence AV s a column-vector of Mλ U(H, O E ) and we conclude by (a). Defnton Gven two E-ratonal onts, q X, the level between and q, denoted lev(, q), s P, Q for any two rmtve ntegral lfts P, Q of, q resectvely. When we are gven a referred E-ratonal ont X, the deth of an E-ratonal ont X s the level between and. By Lemma ths s well-defned when O E s a rncal deal doman. The roxmal cus comlex of level n, denoted C n (Γ), s the comlex whose vertces are cus onts of Γ, wth an edge connectng vertces, q whenever lev(, q) n, and a trangle for each trle of dstnct edges. Levels gve a convenent way to dstngush orbts of edges and trangles n the coverng comlex, by the followng observaton whch follows from Lemmas and : Lemma 3 If O E s a rncal deal doman, for any two E-ratonal onts, q X and γ U(H, O E ), lev(γ, γq) = lev(, q). More mortantly, levels allow us to fnd the otmal heght u of a horoshere H u = B u based at a referred E-ratonal ont X such that the orbt ΓB u covers X. Ths reles on the followng result, whch follows from Corollary 4.6 of [KP]: Lemma 4 There exsts a decreasng functon u : N R such that, for any E-ratonal ont X wth deth n, and any (ntegral) A Γ satsfyng A ( ) =, the set H u A (H u ) s emty f and only f u > u(n). In fact we wll see n Corollary below that the functon u s gven by u(n) = n. Defnton The coverng deth of Γ s the unque n N such that n+ < u cov n, where u cov denotes the maxmal heght such that ΓB u cov covers X..3 Reducton modulo the vertex stablzer Γ We choose a referred cus ont X of Γ (n general we wll tae = π([,,..., ] T ) n the Segel model, see secton 3), and consder the cus stablzer Γ = Stab Γ ( ). Snce Γ s a lattce, t s well nown that Γ acts cocomactly on all horosheres based at. Let n denote the coverng deth of Γ and u cov the corresondng coverng heght, so that the Γ-translates of the horoball B u cover X, and let D H u be a comact fundamental doman for the acton of Γ on H u X \ { }. In ractce we wll choose D to be an affnely convex olytoe n Hesenberg coordnates (see secton 3). Assume that we are gven a fnte resentaton Γ = S R. Then we may reduce the rocedure n Macbeath s theorem to fntely many addtonal generators and relatons as follows. Let {,..., } denote the E-ratonal onts wth deth at most n n D, and assume for smlcty that they are ordered n such a way that the frst r of them form a system of reresentatves under the acton of Γ. Assume moreover that we have found for each =,..., r an element A Γ such that A ( ) = (ths s ossble n rncle snce Γ s assumed to have a sngle cus). Generators: The grou Γ s generated by {S, A,..., A r }. Ths follows easly from art () of Macbeath s theorem and Lemma 4, as any E-ratonal ont of X wth deth at most n s n the Γ -orbt of one of 4

5 ,..., r. Note that wth the notaton from the Theorem, we are usng the oen set V = B u = B to cover X, and E(B) = {γ Γ B γb } = Γ {A,..., A r }Γ = {γ A γ γ, γ Γ, =,..., r}. Indeed, by Lemma 4, B γb f and only f γ s ether or an E-ratonal ont of deth at most n, whch s a Γ -translate of one of,..., r. Relatons: We now rehrase art () of Macbeath s theorem n ths context. Let γ, γ E(B) satsfy B γb γγ B, and frst assume that γ and γγ. After conugatng by an element of Γ we may asume that γ = A, γ = γ A γ, γγ = γ A 3 γ 4 for some,, {,..., r} and γ,..., γ 4 Γ. The corresondng relaton γγ = γ γ s then: A γ A γ = γ A 3 γ. 4 Tang the mage of under both sdes of ths relaton gves: A (γ ) = γ 3. In ractce ths s how we wll detect the relatons, fndng whch onts of deth at most n are sent to onts of deth at most n by the generators A. One then recovers the relaton as follows. For each trle (,, ) for whch there exst γ, γ 3 Γ such that A (γ ) = γ 3, we obtan a relaton R,, by dentfyng the element (γ3 ) A γ A Γ as a word n the generators S. Now assume that one of γ, γγ s but not both (as the relatons n Γ have already been consdered). The corresondng relaton can be obtaned as above, usng the ont = wth corresondng grou element A = Id. Summarzng the above dscusson gves: Lemma 5 Wth the above notaton, Γ admts the resentaton Γ = S, A,..., A r R, R,,..4 The method n ractce We now gve an outlne of the method we use to aly Macbeath s theorem: () Fnd an exlct (affne) fundamental doman D X H u for the acton of Γ = Stab Γ ( ), and a resentaton Γ = S R. () Fnd the coverng deth n of Γ. Consder the corresondng coverng comlex C(ΓB u cov) C n (Γ). (3) Fnd all E-ratonal onts {,..., } n D wth deth at most n (and denote = ). (4) For each of the r Γ -orbt of onts, fnd an exlct A Γ such that A ( ) =. (5) For each trle (,, ) for whch there exst γ, γ 3 Γ such that A (γ ) = γ 3, we obtan a relaton R,, by dentfyng the element (γ3 ) A γ A Γ as a word n the generators S. Then by Macbeath s theorem and Lemma 5, Γ = S, A,..., A r R, R,,. In order to avod tedous reetton of smlar arguments or straghtforward comutatons, we wll only gve one detaled roof for each ste for the Pcard modular grous, but we wll gve detaled arguments for the quaternonc lattce when they are substantally dfferent. We wll usually choose the most dffcult case, or the most nstructve f the varous cases are of smlar dffculty. Stes () and (5) are routne and we ust state the results (excet ste () for the quaternonc lattce, whch we cover n detal n Lemmas 5 and 6). We gve a detaled argument and roof for ste () for the Pcard modular grou Γ(7) n Lemma 3, and for ste (3) for the same grou and deth n Lemma 4. There seems to be no general strategy for ste (4); we fnd all relevant matrces n ths aer by combnng two trcs, whch lucly cover all the cases we need. The frst trc s to use stablzers of vertcal comlex lne n the Hesenberg grou: t s easy to fnd such a matrx when t stablzes the vertcal axs, then we carry over to other vertcal lnes by conugatng by a horzontal translaton. The second trc s to ht all relevant ntegral onts by the grou elements that we already now, and see f we land n the Γ -orbt of the ont we are tryng to reach..5 A toy examle: Γ = PSL(, Z) In order to llustrate the method, we now go through ts stes for Γ = PSL(, Z) exactly as we wll for the more comlcated Pcard and Hurwtz modular grous. The results are ether well-nown or elementary and we state them wthout roof. 5

6 Presentaton and fundamental doman for the cus stabllzer Γ : The cus stablzer Γ has resentaton T ; a fundamental doman for ts acton on H R \ { } R s D = [, ]. Concretely we use the followng generator for Γ : [ ] T = Coverng deth and Q-ratonal onts n D : The coverng deth of PSL(, Z) s. The Q-ratonal onts of deth n D are and, both n the same Γ -orbt. An ntegral lft of s = [, ] T ; we denote = T. Generators: The followng element A Γ mas the ont = [, ] T to : [ ] A = I = Relatons: We lst n Table the relatons obtaned for PSL(, Z) by alyng generators to onts of deth at most as descrbed n art (5) of secton.4. The second relaton s obtaned by followng the corresondng cycle of onts, whch gves I T I T I Γ. The latter element s comuted to be T, gvng the relaton (I T ) 3 = Id. Image of vertex Cycle of onts Relaton I = I = T I I I (T I ) (T I ) I = Id (I T ) 3 = Id Table : Acton of generators on vertces for PSL(, Z) 3 Horoshere ntersectons Our man reference for ths secton s [KP]. We wll use the Segel model of hyerbolc sace H n K (wth K = R, C, H), whch s the roectve model (as descrbed n Secton.) assocated to the Hermtan form on K n+ gven by Z, W = W JZ wth: J = I n Then hyerbolc sace H n K can be arametrzed by Kn Im K R + as follows, denotng as before by π the roectvzaton ma: H n K = {π(ψ(ζ, v, u) ζ Kn, v Im K, u R + )}, where: ψ(ζ, v, u) = ( ζ u + v)/ ζ () Wth ths arametrzaton the boundary at nfnty H n K corresonds to the one-ont comactfcaton: { π(ψ(ζ, v, ) ζ K n, v Im K } { } where = π((,,..., ) T ). The coordnates (ζ, v, u) K n Im K R + are called the horoshercal coordnates of the ont π(ψ(ζ, v, u) H n K. Defnton 3 For a fxed u R +, the level set H u = {π(ψ(ζ, v, u ) ζ K n, v Im K} s called the horoshere at heght u based at, and B u = {π(ψ(ζ, v, u) ζ K n, v Im K, u > u } s called the horoball at heght u based at. 6

7 The unctured boundary H n K \ { } s then naturally dentfed to the generalzed Hesenberg grou Hes(K, n), defned as the set K n Im K equed wth the grou law: (ζ, v )(ζ, v ) = (ζ + ζ, v + v + Im (ζ ζ )) where denotes the usual Eucldean dot-roduct on K n. Ths s the classcal 3-dmensonal Hesenberg grou when K = C and n =. The dentfcaton of H n K \ { } wth Hes(K, n) s gven by the smly-transtve acton of Hes(K, n) on H n K \ { }, where the element (ζ, v ) Hes(K, n) acts on the vector ψ(ζ, v, ) by left-multlcaton by the followng Hesenberg translaton matrx n U(n,, K): ζ ( ζ + v )/ T (ζ,v ) = I n ζ (3) Gven an element U U(n, K), the Hesenberg rotaton by U s gven by the followng matrx: R U = U (4) There s an addtonal class of sometres fxng when K = H, comng from the acton of dagonal matrces whch s non-trval n the non-commutatve case. Recall that our conventon s that matrces act on vectors on the left, and scalars act on vectors on the rght. Then, for any unt quaternon q H, the dagonal matrx C q = q Id acts by the sometry of hyerbolc sace gven by conugatng horoshercal coordnates (the result of multlyng the vector form () by q on the left, then normalzng by q on the rght): C q : (ζ, v, u) (qζq, qvq, u) (5) For ths reason, when K = H the relevant roectvzaton of U(n,, H) actng on H n H s PU(n,, H) = U(n,, H)/{±Id} rather than U(n,, H)/U(). Hesenberg translatons and rotatons, as well as conugaton by unt quaternons, reserve the followng dstance functon on Hes(K, n), called the Cygan metrc, defned for (ζ, v ), (ζ, v ) Hes(K, n) by: d C ((ζ, v ), (ζ, v )) = ζ ζ 4 + v v Im (ζ ζ ) /4 = ψ(ζ, v, ), ψ(ζ, v, ) / (7) Ths n fact the restrcton to H n K \{ } of an ncomlete dstance functon on Hn K \{ } called the extended Cygan metrc (see [KP]), defned for (ζ, v, u ), (ζ, v, u ) K n Im K R H n K \ { } by: d XC ((ζ, v, u ), (ζ, v, u )) = ( ζ ζ + u u ) + v v Im (ζ ζ ) /4 (8) = ψ(ζ, v, u ), ψ(ζ, v, u ) / (9) We defne Cygan sheres, Cygan balls, extended Cygan sheres and extended Cygan balls n the usual way relatve to these dstance functons. When we aly Macbeath s theorem we argue that the mages under Γ of the horoball B u based at at a certan heght u > cover X, or equvalently cover the horoshere H u = B u. The followng result, whch follows from Prooston 4.3 of [KP], allows us to control the traces on H u of these mages only n terms of Cygan sheres deendng only on arthmetc data. Lemma 6 Let A = (a, ) U(n,, K) such that A( ), S the extended Cygan shere wth center ( ) and radus / a n,, and H u the horoshere based at at heght u >. Then H u A(H u ) = H u S. Corollary Let E be a number feld such that O E s a rncal deal doman, X an E-ratonal ont wth deth n and A U(H, O E ) satsfyng A ( ) =. Then H u (H u ) = H u S, where S s the Cygan shere centered at wth radus ( ) 4 /4. n In artcular: Hu (H u ) = u > u(n) = n. 7 (6)

8 Proof. Snce A ( ) = and e = (,,..., ) T s a lft of, the frst column vector of A s a lft P of, and snce A U(H, O E ) t s an ntegral lft. In fact by Lemma t s a rmtve lft, therefore the deth of s P, e = a n,, denotng as above A = (a, ), and the result follows from Lemma 6. The second art of the statement follows by usng ths radus n the formula for the extended Cygan metrc, Equaton 8. We wll also use the followng observaton, whch s Lemma of [FFP], n our coverng arguments: Lemma 7 Extended Cygan balls are affnely convex n horoshercal coordnates. Fnally, when consderng the acton of a dscrete subgrou Γ of Isom( X) (relatve to the Cygan metrc) t s convenent to consder ts vertcal and horzontal comonents defned as follows (see [FP] for the case K = C and n = ). The homomorhsm Π : Hes(K, n) K n gven by roecton to the frst factor n the decomoston Hes(K, n) K n Im K nduces a short exact sequence: Isom(Im K) Isom(Hes(K, n)) Π Isom(K n ), () where the sometres of Im K and K n are relatve to the Eucldean metrc. Denotng Γ v = Γ Isom(Im K) and Γ h = Π (Γ ) ths gves the short exact sequence: 4 Pcard modular grous Γ v Γ Π Γ h. () In ths secton we use the method descrbed n Secton.4 to comute the followng resentatons for the Pcard modular grous Γ(d) = PU(,, O d ) wth d =, 3, 7. We wll denote τ = + d when d 3 (mod 4), so that O d = Z[τ]. [[T, T τ ], T ], [[T, T τ ], T τ ], [[T, T τ ], R], PU(,, O 3 ) = T, T τ, R, I R T τ R = T, R T R = T Tτ, R 6,. () I, [R, I], ([T, T τ ]I) 6, T Tτ [T, I]T τ T = I[T, T τ ] PU(,, O ) = T, T τ, T v, R, I [T τ, T ] = Tv 4, [T v, T ], [T v, T τ ], [T v, R], R 4, I, [R, I], RT R = Tτ T Tv 4, RT τ R = T τ T Tv, [I, T ], (IT v ) 3 = R, [I, T τ ] = T τ IR, (T v IR Tv I), (3) ITv T τ IRT Tv = T Tτ IT τ R T v I, (ITv T τ IRT Tv ) = R T T τ Tv 3 PU(,, O 7 ) = T, T τ, T v, R, I, I [Tτ, T ] = T v, [T v, T ], [T v, T τ ], [T v, R], R, (RT τ ), (RT ) = T v, I, I, [R, I ], [R, I I T T τ ], [R, I I T T τ ] = T v I I T τ T I I T τ T T v, [R, I I T T τ ] = T v T I T I Tτ I RI, [I, Tv T τ T ] = T I I I T, R[R, I I T T τ ] = T I T v T I T T I = T T τ RT I T I T I The acton of Γ (d) = Stab Γ(d) ( ) on H C s well understood for all d, see [FP] for d = 3, [FFP] for d = and Secton 5.3 of [PW] for all other values (usng unublshed notes of Falbel-Francscs-Parer). We wll refer to these aers for resentatons and fundamental domans for Γ (d) whch we state n Lemmas 8, and. v R, (4) 8

9 Fgure : Coverng the rsm D (3) by Cygan balls of deth 4. The Esensten-Pcard modular grou Γ(3) = PU(,, O 3 ) Presentaton and fundamental doman for the cus stablzer Γ (3): Lemma 8. The cus stablzer Γ (3) admts the followng resentaton: Γ (3) = T, T τ, R [[T, T τ ], T ], [[T, T τ ], T τ ], [[T, T τ ], R], R T τ R = T, R T R = T T τ, R 6.. Let D (3) H C be the affne convex hull of the onts wth horoshercal coordnates (, ), (, ), ( τ+ 3, ), (, 3), (, 3), ( τ+ 3, 3). Then D (3) s a fundamental doman for Γ (3) actng on H C \ { }. Concretely, we use the followng generators for Γ (3) (recall that τ = + 3 T = T (, 3) = τ T τ = T (τ, 3) = τ τ τ ) R = τ Coverng deth and Q[ 3]-ratonal onts n D (3): We denote B ((z, t), r) the oen extended Cygan ball centered at = (z, t) H C wth radus r (see Equaton 8 for the defnton of the extended Cygan metrc). Recall that u(n) = n s the heght at whch balls of deth n aear, n the sense of Corollary. Lemma 9 Let u = u(5) + ε =.895 and H u the horoshere of heght u based at. Then the rsm D (3) {u} s covered by the ntersectons wth H u of the followng extended Cygan balls of deth : B ( (, ), ), B ( (, 3), ) and B ( (, 3), ). We omt the roof, whch s smlar to the roof of Lemma 3 but much smler; see Fgure 4.. 9

10 Note that n order to cover D (3) {u} we only need balls of deth, n artcular none of deths 3 or 4 even though they are resent at the heght u = u(5) + ε whch we consder. It s however necessary to ass to ths heght, as we have observed exermentally that at heght u = u(4) + ε balls of deth at most 3 do not cover D (3) {u} (there are no Q[ 3]-ratonal onts of deth ). Corollary The coverng deth of Γ(3) s at most 4. By nsecton, we see that the Q[ 3]-ratonal onts n D (3) wth deth at most 4 are, n horoshercal coordnates: Deth : (, ), (, 3) and (, 3), all n the same Γ (3)-orbt; Deth 3: (, 3 3), (, 5 3 3) n one Γ (3)-orbt, and (, 4 3 3), (, 3 3) n the other; Deth 4: (, 3), (, ) and (, 3), all n the same Γ (3)-orbt. Integral lfts of reresentatves of Γ (3)-orbts of these onts are: = 3, = 3, = 3 4 = 3 3 Generators: The followng elements A α Γ(3) ma the ont = [,, ] T to the corresondng α as above (for α = ; 3, ; 3, ; 4) : 3 A = I = A 3, = A 3 = 3 A 3, = 3 A 4 = 3 3 Relatons: We lst n Table the relatons obtaned for Γ(3) by alyng generators to onts of deth at most 4 as descrbed n art (5) of secton.4. By successvely elmnatng A 4 then A 3 we obtan resentaton (). The detals are straghtforward and left to the reader. 4. The Gauss-Pcard modular grou Γ() = PU(,, O ) Presentaton and fundamental doman for the cus stablzer Γ (): Lemma. The cus stablzer Γ () admts the followng resentaton: Γ () = T, T τ, T v, R [T τ, T ] = Tv 4, [T v, T ], [T v, T τ ], [T v, R], R 4, RT R = Tτ T Tv 4, RT τ R = T τ T Tv.. Let D () H C be the affne convex hull of the onts wth horoshercal coordnates (, ), (, ), (τ, ), (, ), (, ), (τ, ). Then D () s a fundamental doman for Γ () actng on H C \ { }. Concretely, we use the followng generators for Γ () (denotng τ = + ): τ T = T (,) = T τ = T (τ,) = τ T v = T (,) = R =

11 Image of vertex Cycle of onts Relaton I = A 4 4 = R = R 3, = 3, R 3, = 3, I I A4 A 4 4 I R I R 3, 3 3, A3 A 3 R A 3, 3 3, I (, 3) = 3, TvI (, 3) TvI A 3 3, I (, 3) = T τ TI (, 3) T τ T vi I I 3, = A3 T vi I 3, 4 I 3, = 4 A 3 T vi 3, 4 I 4 = 3, A4 T vi 4 3, A 3 = 4 A 3 4 = T v A 3 3, = 3, A 4 = 3, A 4 3, = T v I A 3 4 A4 4 T v A3 3 4 I I T τ I = Id A 4 = Id [R, I ] = Id [R, A 3 ] = Id [R, A3 ] = Id A3 = R 3 (T v I ) T v I T I = T T T A3 = I T v v I R 3 T v A 4 = T v I 3 R3 A 4 = I T v A 3 R 3 A 4 = A 3 I T v R 3 A4 = 3 T vi R 3 A3 A 3 A 3, 3 3, A 3 3 = R 3 I A 4 A 3 3, A3 3, T v A4 I A4 = 3 R3 T v I A4 = A 3 R 3 I T v τ Table : Acton of generators on vertces for d = 3 Coverng deth and Q[]-ratonal onts n D (): We denote B ((z, t), r) the oen extended Cygan ball centered at = (z, t) H C wth radus r (see Equaton 8 for the defnton of the extended Cygan metrc). Recall that u(n) = n s the heght at whch balls of deth n aear, n the sense of Corollary. Lemma Let u = u(5)+ε =.895 and H u the horoshere of heght u based at. Then the rsm D () {u} s covered by the ntersectons wth H u of the followng extended Cygan balls: (deth ) B ( (, ), ), B ( (, ), ), B ( (τ, ), ), B ( (τ, ), ), B ( (, 7), ), (deth ) B ( (, ), 4 ). We omt the roof, whch s smlar to the roof of Lemma 3 but smler; see Fgure 4.. Note that n order to cover D () {u} we only need balls of deth at most, n artcular none of deth 4 even though they are resent at the heght u = u(5) + ε whch we consder. It s however necessary to ass to ths heght, as we have observed exermentally that at heght u = u(4) + ε balls of deth at most do not cover D () {u} (there are no Q[]-ratonal onts of deth 3). Corollary 3 The coverng deth of Γ() s at most 4. By nsecton, we see that the Q[]-ratonal onts n D () wth deth at most 4 are, n horoshercal coordnates: Deth : (, ), (, ), (τ, ) and (τ, ), all n the same Γ ()-orbt; Deth : (, )

12 Fgure : Coverng the rsm D () by Cygan balls of deth and Deth 4: (, ), (τ, ) n one Γ ()-orbt and (, ), (, ) n the other. Integral lfts of reresentatves of Γ ()-orbts of these onts are: = = + 4, = 4, = + Generators: The followng elements A α Γ() ma the ont = [,, ] T to the corresondng α as above (for α = ; ; 4, ; 4, ) : + A = I = A = A 4, = A 4, = 3 Relatons: We lst n Table 3 the relatons obtaned for Γ() by alyng generators to onts of deth at most 4 as descrbed n art (5) of secton.4. By successvely elmnatng A 4,, A 4, then A we obtan resentaton (3). The detals are straghtforward and left to the reader. 4.3 The Pcard modular grou Γ(7) = PU(,, O 7 ) Presentaton and fundamental doman for the cus stablzer Γ (7): Lemma. The cus stablzer Γ (7) admts the followng resentaton: Γ (7) = T, T τ, T v, R [T τ, T ] = T v, [T v, T ], [T v, T τ ], [T v, R], (RT τ ), (RT ) = T v, R. Let D (7) H C be the affne convex hull of the onts wth horoshercal coordnates (, ), (, 7), (τ, ), (, 7), (, 3 7), (τ, 7). Then D (7) s a fundamental doman for Γ (7) actng on H C \{ }.

13 Image of vertex Cycle of onts Relaton I = I I A 4, 4, = A4, 4, A 4, A 4, 4, = A4, 4, A 4, R = I R I R 4, = 4, A4, R 4, 4, 4, I (, ) = TvI (, ) TvI I I (τ, ) = (τ, ) I (τ, ) = T τ T A Tτ I (τ, ) I I T τ T I = T τ A (τ, ) (τ, ) I = Id A 4, = Id A 4, = Id [R, I ] = Id [R, A 4, ] = Id (I T v ) 3 = R (Tτ I) [I, T τ ] = T τ I R 3 (TvTτ I) T τ T vi I I 4, = Tv A4, T v I I 4, I 4, = T A4, 4, A = Tτ T Tv 4 A4, T v I I 4, A = A A A 4, = T τ T T 4 v A4, 4, A = T T τ I T v T τ I R A = I Tv T τ I RT Tv A4, = I Tv I RTv T I I A4, = [I, T ] T 4 v T T τ A T A A 4, = T v A4, v A 4, A 4, = T v A 4, = A 4, = T I T v A4, I A A 4, I A4, = I Tv I RTv A = R T T τ Tv 3 A 4, = T τ T T 4 v A R T A4, = T va T Tτ R A4, = T v I R T v I A 4, = (τ, ) A T va 4, (T τi ) (τ, ) A 4, = A R Tτ T T A 4, I A4, = [T, I ] Tv A 4, = Tv A R T τ T Table 3: Acton of generators on vertces for d = Concretely, we use the followng generators for Γ (7) (denotng τ = + 7 ): τ τ T = T (, 7) = T τ = T (τ,) = τ T v = T (, 7) = 7 R = Coverng deth and Q[ 7]-ratonal onts n D (7): We denote B ((z, t), r) the oen extended Cygan ball centered at = (z, t) H C wth radus r (see Equaton 8 for the defnton of the extended Cygan metrc). Recall that u(n) = n s the heght at whch balls of deth n aear, n the sense of Corollary. Lemma 3 Let u = u(8) + ε =.77 and H u the horoshere of heght u based at. Then the rsm D (7) {u} s covered by the ntersectons wth H u of the followng extended Cygan balls: (deth ) B ( (, ), ), B ( (, 7), ), B ( (τ, ), ), B ( (τ, 7), ), B ( (, 7), ), (deth ) B ( (τ/, 3 7/), 4 ), B ( ((τ + )/, 7), 4 ) 3

14 Fgure 3: Coverng the rsm D (7) by Cygan balls of deth, and 4 (deth 4) B ( (τ, 7), ). Proof. Fgure 4.3 shows the rsm D (7) and the relevant Cygan balls. We rove the result by dssectng the rsm D (7) {u} nto affne olyhedra, each of whch les n one of the extended Cygan balls. Ths s remnscent of the roof of Prooston 5. of [Zh]. Consder the followng onts of H C, n horoshercal coordnates (see Fgure 4.3): q = (.65τ,.8) q = (τ, ) q 3 = (τ, 3.3) q 4 = ( τ, 3.4) q 5 = (.7 +.3τ, 4.) q 6 = (, 4.3) q 7 = ((τ + )/,.5) q 8 = (τ,.5) q 9 = ((τ + )/, ) q = (, ) q = (τ, 4) q = ((τ + )/, 7) q 3 = (τ/, 4) q 4 = (τ/, 7) q 5 = (, 3.5) q 6 = (.3τ, 3) q 7 = (τ/, ) q 8 = (,.7) q 9 = (τ/, ) Denotng Hull(S) the affne hull (n horoshercal coordnates) of a subset S H u H C {u}, we clam that the followng affnely convex eces of D (3) {u} are each contaned n the corresondng (oen) extended Cygan shere: C = Hull ((, ), (, ), q 9, q, q 7, q 8, q 9 ) B ( (, ), ) C = Hull ( (, 7), (, 7), q 6, q, q 3, q 4, q 5 ) B ( (, 7), ) C 3 = Hull ((τ, ), q 7, q 8, q 9, q 7, q 9 ) B ( (τ, ), ) C 4 = Hull ( (τ, 7), q, q, q 3, q 4 ) B ( (τ, 7), ) C 5 = Hull ( (, 7), q 6, q 7, q, q 5, q 6, q 7, q 8 ) B ( (, 7), ) C 6 = Hull (q, q 3, q 4, q 5, q, q, q 3, q 5, q 6 ) B ( (τ/, 3 7/), 4 ) C 7 = Hull (q, q, q 4, q 5, q 6, q 7, q 8, q 6, q 7 ) B ( ((τ + )/, 7), 4 ) C 8 = Hull (q, q, q 3, q 4 ) B ( (τ, 7), ) 4

15 To verfy each of these clams, we chec numercally that each of the vertces ndeed belongs to the ball n queston usng Equaton (8), then extend to the whole affne covex hull by Lemma 7. For examle, the ont q = (.65τ,.8) ndeed belongs to B ( (τ/, 3 7/), 4 ), B ( ((τ + )/, 7), 4 ) and B ( (τ, 7), ) because: d XC ( (.65τ,.8, u), (τ/, 3 7/, ) ).79 < 4.89 d XC ( (.65τ,.8, u), ((τ + )/, 7, ) ).7 < 4.89 d XC ( (.65τ,.8, u), (τ, 7, ) ).98 < The result then follows as the rsm D (7) {u} s the unon of the affnely convex eces C,..., C 8, see Fgure 4.3. Corollary 4 The coverng deth of Γ(7) s at most 7. Note that n the above coverng argument we have only needed balls of deth at most 4 (n artcular none of deth 7) even though they are resent at the heght u = u 8 + ε whch we consder. It s however necessary to ass to ths heght, as we have observed exermentally that at heght u = u 7 + ε balls of deth at most 4 do not cover D (7) {u} (there are no Q[ 7]-ratonal onts of deths 5 or 6). By nsecton, we see that the Q[ 7]-ratonal onts n D (7) wth deth at most 7 are the followng, n horoshercal coordnates. We gve n Lemma 4 below a detaled ustfcaton for the onts of deth. Deth : (, ), (, 7), (τ, ), (τ, 7) and (, 7), all n the same Γ (7)-orbt; Deth : ( τ, 3 7) n one Γ (7)-orbt and ( τ+, 7) n the other; Deth 4: (, 7), (τ, 7), (, ), (, 7) n one Γ (7)-orbt, ( τ, 7) n a second and ( τ+ τ+, ), (, 7) n the thrd; Deth 7: for each =,..., 6, there s a Γ (7)-orbt contanng the 3 onts (, 7 7), (τ, 7 7), 7) (wth + 7 taen mod 7). (, +7 7 Integral lfts of reresentatves of Γ (7)-orbts of these onts are: = 7, = 7, = 7, = τ τ 7, = 7,3 = τ τ 3 7 4, = 7 7,4 = 4 7 4, = 7,5 = τ τ 5 7 4,3 = 7,6 = Lemma 4 The Q[ 7]-ratonal onts n D (7) wth deth are exactly ( τ, 3 7) and ( τ+, 7). Proof. We llustrate the general rocedure for fndng onts of deth n, then secalze to the resent case. The deths at whch there wll be E-ratonal onts are the natural numbers n such that z = n has a soluton z O d. For d = 7, the frst few n s are,, 4, 7,.... Begn by assumng we have found all onts of deth less than n. In ths case those are ust the onts at deth.. Fnd all q = x + yτ O d (u to multlcaton by a unt) wth q = n. There are two ossbltes: ether q = τ := + 7 or q = τ. τ

16 (, 7) q 4 (, 7) (, 7) (τ, 7) (, 7) q 4 q q 6 q 3 4 q 4 6 q 5 q 6 q 5 6 q 3 q 4 5 q 6 q q q 8 q 8 q 7 5 q q 7 q 3 3 (, ) (, ) (, ) q 9 (τ, ) q 9 q 9 3 (, ) Fgure 4: Affne cell decomoston of the rsm D (7) 6

17 . Consder the standard lft of a ont of H K \ { }: P = z +t z Note that t = b 7 for some b Q. The calculaton s more transarent f we rewrte the frst coordnate: z + t = z b For P to be the vector form of an E-ratonal ont of deth, t must satsfy the followng: + bτ (a) P does not have deth (n general, P does not have deth less than n). coordnates of P are not n O 7. (b) P q s coordnates must be n O 7 for some q from ste (). In other words, the Next, we do some calculatons to mae sure (b) s satsfed. 3. Fnd all z n the roecton to C of D such that zq O 7. If q = τ we can have z =,, τ, or +τ. If q = τ, we can have z =,, τ, τ. 4. For each ossble z, fnd z and comute ( ) and z b + bτ q O 7. +τ ( ) z b + bτ q. Use ths to lst all b s such that (z, b) D ( ) z z q z b + bτ q b s ont(s) n horo. coords τ b + bτ, (, ), (, 7) τ 3b b τ, (, ), (, 7) τ b + b τ (, 7) 3b τ +b τ (, 7) τ τ b b, (, ), (, 7) 3b τ +b τ, (, ), (, 7) τ b + b τ ( +τ, 7 ) ( ) τ 6b τ 4 +b 3 τ 4 τ, Get rd of{ all the ones that are level (or from any revous level). What you are left wth are the level ( onts: +τ, 7 ) ( )} τ,, 3 7. Generators: The followng elements A α Γ(7) ma the ont = [,, ] T above (for α = ;, ;, ; 4, ; 4, ; 4.3; 7, ;...; 7, 6) : to the corresondng α as 7

18 A = I = A 4, = A 7, = A 7,4 = A, = A 4, = A 7, = τ 5 5 τ 7 7 τ τ τ τ τ τ A 7,5 = A, = A 4,3 = A 7,3 = τ τ τ τ τ τ τ τ + 3 τ τ A 7,6 = Relatons: We lst n Table 4 the relatons obtaned for Γ(7) by alyng generators to onts of deth at most 7 as descrbed n art (5) of secton.4. By successvely elmnatng A,, A 7,, A 7,4, A 7,6, A 7,, A 7,3, A 7,5, A,, A 4,3 then A 4,, and settng I = A 4, we obtan resentaton (4). The detals are straghtforward and left to the reader. 5 The Hurwtz quaternon modular grou PU(,, H) In ths secton we use the method descrbed n Secton.4 to comute the followng resentaton for the Hurwtz modular grou Γ(H) = PU(,, H), sometmes denoted PS(,, H). Recall that the Hurwtz nteger rng s H = Z[,,, σ] H, where we denote σ = +++. The generators S and relatons R for Γ (H) are defned n Lemma 5. PU(,, H) = S, I [Cσ [Cσ R, I, [I, R ], [I, R σ ], [I, C ], [I, C σ ], [ITv IR C, R σ ], (IT v v ) 3 R C σ, R ], [Cσ (I ) 3 R C σ, R σ ], (IT ) 3 R C σ, C σ ], (ITv IR C ), (T v I) 3 = R C, (T v T v I) 4 = R, (T v T v T v I) 6, I (IT v ) = (T v I ) R, [(T v T v I), Cσ (IT T T ) 3 R C σ], IT Tσ IT σ I = T T T T v v IT T T v v Tv I = T σ IT v T v IT T σ C T v v v Cσ, (5) Presentaton and coarse fundamental doman for Γ (H): In ths secton we study the acton of Γ (H) on H H \ { } H Im H. We use the notaton of Equatons (3), (4) and (5), namely T (ζ,v), R w, C w resectvely denote Hesenberg translaton by (ζ, v), Hesenberg rotaton by w and conugaton by w. Note that for any unt quaternon w H and urely magnary q H, Γ (H) contans the followng Hesenberg translatons: T w = T (w,++) = w +++ w, T vq = T (,q) = q (6) Lemma 5 Γ (H) admts the followng resentaton: Γ (H) = S R = R, R σ, C, C σ, T, T, T, T, T σ, T v, T v, T v A, A, A 3, A 4, A 5 8

19 Image of vertex Cycle of onts Relaton I = I I A 4, 4, = A4, 4, A 4, A 4, 4, = A4, 4, A 4, A 4,3 4,3 = A4,3 4,3 A 4,3 R = I R I I, = T 4,3 A,, T I 4,3 I 4, = T v 7,5 A4, 4, T vi 7,5 4,3 7,5 I 4, = T, A4, T vt I, 4,, I 4,3 = T τ T I I T τ T T v 4,3 I 7, = I I T v 7, I 7, = 4, A4, I T v 4, 7, I = Id A 4, = Id A 4, = Id A 4,3 = Id [R, I ] = Id A 4,3 = T I A, T A 7,5 = RT v I A 4, 7, 4,3 7, 4, = T T τ T v R, T vt I A 4,3 = I T τ T T v I T A 7, = I Tv I R A 7, = I T v A 4, I (, 7) = 7,6 TvI (, 7) TvI 7,6 7,6 A 7,6 = (T v I ) I (, 7) =, TI (, 7) TvI,, I (τ, ) = (τ, ) A, = 7,5 A, = 4, A 4, = 7,4 Tτ I (τ, ) I (τ, ) I A, 7,5 I A, 4, (Tτ I) 7,5 4, A, = T v I T I T v T (I T τ R) 3 = Id A 7,5 = A, I T Tv R A 4, = A, I I A 4, 7,4 7,4 A 7,4 = A 4, I A 4, 7,3 = T v A7,3 T v A4, I 7,3 A 4, =, A 4, 7, = T τ T, A7, I A 4,, 7, A 4,3 = T τ T I A 7, =, A7,3 = 4, T vi R A 4, = A, I T T τ A 4,,, T T τ T va 4,3 I I A 7, I A 7, 4, = A4, 4, A 7, I A 7, 7,3 = A7,3 7,3 A 7, A 7, 7,4 = 7, A7,4 A 7, 7, 7,4 7, A 7,3 = 4, I A 7,3 4, 4, A 7,3 4, = T, A4, T A 7,3, 4,, τ T A 7, = 4, T τ T A, T τ T T v A4,3 = Tv T τ T I T Tτ T v I A7, = I RT v I A7, = I Tv 4, A7,3 = 7, R A 7,4 = 7, A 7,R A 7,3 = A 4, RT v I A 7,3 = T A, Tτ T T v 4, A 7,3 7, = 7,4 A7, A 7,3 7,4 7, 7,4 A 7,4 = A 7,3 A 7, A 7,3 7, = A7, 7, A 7,3 A 7,4 = 4, A 7,5, = T v T 4,3 A,, I A 7,4 4, T T v A7,5 4, 4,3 A 7,5 4, = T v A4, T v A7,5 I 4, A7,3 = 7, R A 7,4 = A 4, I 4,3 A 7,5 = T v T A 4,3 RTτ T, A7,5 = T v I R 4, A 7,5 7,3 = 7,6 A7,3 A 7,5 7,6 7,3 7,6 A 7,6 = A 7,5 A 7,3 A 7,5 7,4 = A7,4 7,4 A 7,5 A 7,6 = T v A7,5 = RT v 7,4 I T v A7,6 I A7,6 = (T v I ) A 7,6 7, = A7, 7, A 7,6 A 7,6 7, = 7,5 A7, A 7,6 7,5 7, 7,5 9 Table 4: Acton of generators on vertces for d = 7 A7,6 = RT v 7, A 7,6 = A 7,5 R 7,

20 where the A n are the followng sets of relatons: A = {R 4, R R 3 σ, (R R σ ) 3, (R R σ ) R A = {C, C 3 σ, (C C σ ) 3 } Rσ } { [Tvw, T ], [T, T w ] = Tv A 3 = w, [T, T σ ] = [T w, Tŵ] = Tv wŵ, [T w, T σ ] = T vw T vŵ T vwŵ, where T runs over T, T, T, T, T σ, T v, T v, T v, w runs over,,, and î =, ĵ =, ˆ =. A 4 = { C R C = R, C R σ C = R R 4 σr, C σ R C σ } = R σ R R σ, C σ R σ C σ = R σ } A 5 = {GT G = E G,T }, where G runs over R.R σ, C, C σ, T runs over T, T, T, T, T σ, T v, T v, T v, and E G,T column and T row of the table below: s the entry n the G R R σ C C σ T T T σ T T T T v T v T v T T T σt v Tv T T T v T T T T T T σ T v T v T T v T T T T v T v T v T T T σ T v Tv T T v T T σ Tσ T T T v T T σ T T T σt v T σ T v T v T v T v T v T v T v T v T v T v T v T v T v Proof. To obtan a resentaton for Γ (H), we dentfy 3 of ts subgrous, observe that some of them normalze each other, and buld u the resentaton va a sequence of extensons usng the followng rocedure. Suose that G s a grou wth subgrous N and K where N s normal n G and G s an extenson of N by K. Suose also that we now resentatons for N and K, N = S N R N and K = S K R K. Then G admts the resentaton S N S K R N R K R, where the set R conssts of relatons of the form n = n where runs over all elements of S K, n runs over the elements of S N, and n N s exressed as a word n the generators S N. The three subgrous of Γ (H) we dentfy are the rotaton, conugaton, and translaton subgrous. The rotaton subgrou conssts of all Hesenberg rotatons R w where w s a Hurwtz ntegral unt quaternon. It s somorhc to the bnary tetrahedral grou, whch has order 4 (see e.g. [CS]). It admts the resentaton R, R σ A The conugaton subgrou conssts of all conugatons by unt quaternons. Elements of ths grou also corresond to Hurwtz unt ntegral quaternons, only C w acts the same as C w. Thus, ths grou s somorhc to the quotent of the bnary tetrahedral grou by, whch s the tetrahedral grou (or the alternatng grou on 4 elements). It admts the resentaton C, C σ A The translaton subgrou conssts of all Hesenberg translatons. It admts the resentaton T, T, T, T, T σ, T v, T v, T v A 3 T v The rotaton subgrou s normalzed by the conugaton subgrou. The extenson of the rotaton subgrou by the conugaton subgrou s a fnte grou of order 88. We obtan the four relatons n A 4 by conugatng R and R σ by C and C σ.

21 The translaton subgrou s normalzed by the rotaton-conugaton subgrou. The conugates of the translaton generators by the rotaton and conugaton generators are lsted n the table. The translatons T are along the left sde, the rotatons/conugatons G are along the to, and the table entry E G,T s the element GT G wrtten as a word n the translaton generators. A 5 contans these relatons. The acton of the translaton subgrou of Γ (H) s relatvely straghtforward, however the fnte grou Γ, (H) fxng and s more comlcated. Namely, the grou of unts n H (whch has order 4 and s generated by say and σ) acts on H H fathfully by rotatons, and wth ernel {±} by conugaton. Together these actons roduce the grou Γ, (H) as a sem-drect roduct of order 88 as descrbed above, and the geometrc comatblty between the acton of ths fnte grou and the translaton subgrou s not clear. For ths reason, we do not mae exlct a fundamental doman for the acton of Γ (H) on H H \ { }, rather we only use a larger subset D (H) whch s geometrcally smler and suffcent for our uroses n the sense that ts Γ (H)-translates cover H H \ { }. Such a set s sometmes called a coarse fundamental doman. Defne D (H) h = Hull(,, +, +, +, +, +, +, +, +, + ) H, and D (H) = D (H) h [, ] 3 H H \ { } H Im H. Lemma 6 The Γ (H)-translates of D (H) cover H H \ { }. Proof. We roceed n several stes, frst consderng the horzontal roectons (as defned n the short exact sequence ()) Γ h = Π (Γ ) of Γ = Γ (H) and of some of ts subgrous. Clam : A fundamental doman for T, T, T, T h n H R 4 s [/, /] 4. Ths s clear as Hesenberg translatons act as ordnary translatons on the horzontal factor. Clam : A fundamental doman for T, T, T, T, R, R, R h n H s [, /] 4 [/, ] [, /] 3 : Ths can be seen by subdvdng the cube [/, /] 4 nto 6 cubes of half sde-length. The grou R, R, R, somorhc to the classcal quaternon grou of order 8, acts on these 6 subcubes wth orbts, and the clam follows from choosng adacent reresentatves of these orbts. Clam 3: A fundamental doman for T, T, T, T, R, R, R, T σ h n H s D (H): h Frst note that D (H) h H s the unon of two half-cubes of edge-length /, D + and D, where D + = Hull(,,,,, +, +, +, +, +, + ) and D s obtaned from D + by negatng the frst coordnate. Each of these s obtaned by cuttng a cube n along a dagonal hyerlane, for examle D + s obtaned by tang half of the cube [, /] 4, cut along the dagonal hyerlane {(x,..., x 4 ) R 4 x = }. Note that ths hyerlane s the equdstant hyerlane between and T σ (); lewse the hyerlane {(x,..., x 4 ) R 4 x + x + x 3 + x 4 = } s the equdstant hyerlane between and T σ T (). Snce Tσ = T T T T, the subgrou T, T, T, T, R, R, R h has ndex n T, T, T, T, R, R, R, T σ h, and the clam follows from the revous clam. Clam 4: A fundamental doman for T, T, T, T, R, R, R, T σ, T v, T v, T v n H H \{ } H Im H s D (H) h [, ] 3 : Ths follows from the revous clam as the addtonal generators T v, T v, T v are vertcal translatons by unts n Hesenberg coordnates. Clam 5: The translates of D (H) h [, ] 3 under T, T, T, T, R, R, R, T σ, T v, T v, T v, C, C, C cover H H \ { }: Recall from (5) that for any unt quaternon w, the matrx C w acts on H H H Im H by conugaton by w on both the horzontal and vertcal factors. Snce each of,, conugate the two others to ther oostes, the clam follows by consderng only the acton on the vertcal factor. Coverng deth and Q[,, ]-ratonal onts n D (H):

22 We denote as before B ((ζ, v), r) the oen extended Cygan ball centered at = (ζ, v) H H wth radus r (see Equaton 8 for the defnton of the extended Cygan metrc d XC ). Recall that u(n) = n s the heght at whch balls of deth n aear, n the sense of Corollary. Lemma 7 Let u = u(6)+ε =.865 and H u the horoshere of heght u based at. Then the rsm D (H) {u} s covered by the ntersectons wth H u of the followng extended Cygan balls of deth : B ( (, ), ), B ( (σ, + + ), ) and B ( (σ, + + ), ). Proof. Recall that D (H) = D (H) h [, ] 3 (, where the base D (H) h H s the unon ) of two half-cubes of edge-length /, D + and D where D + = Hull,,,,, +, +, +, +, +, + and D s obtaned from D + by negatng the frst coordnate. Clam: The half-rsm D + [, ] 3 s contaned n B ( (, ), ) B ( (σ, + + ), ). We searate D + [, ] 3 nto eces by the horzontal hyerlane A = {(x + x + x 3 + x 4, t + t + t 3 ) H Im H t +t +t 3 =.5}. The clam s then verfed by showng that D + [, ] 3 A B ( (, ), ) and D + [, ] 3 A + B ( (σ, + + ), ), where A ± denote the half-saces bounded by A, wth (, ) A and (σ, + +) A +. Ths s done as revously by checng the vertces numercally usng Equaton (8), then extendng the result to ther convex hull by Lemma 7. For the vertces, note that for each of the base vertces of D + there s a 3-cube above t n the vertcal drecton, sanned by,,,, +, +, +, + +. We frst chec usng Equaton (8) that for each of the base vertces, all vertcal vertces above t are n B ( (, ), ) excet for the to vertex (corresondng to + +), and that lewse all of these to vertces are n the other ball B ( (σ, + + ), ). Fnally, we chec that all 33 mdonts between these to vertces and the lower level (sanned by +, + and + ) are contaned n the ntersecton B ( (, ), ) B ( (σ, + + ), ). For examle, the mdont ( + + +, + + /) between (, + ) and (, + + ) satsfes: ( d XC ( + ), + + /, u), (,, ).87 <, ( d XC ( + ), + + /, u), (σ, + +, ).47 <. All other comutatons are smlar. Lewse, relacng D + by D by negatng the frst coordnate gves that the other half-rsm D [, ] 3 s contaned n B ( (, ), ) B ( (σ, + + ), ). Corollary 5 The coverng deth of Γ(H) s at most 5. In fact we susect that the coverng deth of Γ(H) s 4, however the coverng argument s much more delcate at heght u(5) + ε, n artcular the 3 Cygan balls used above do not suffce. By nsecton, we see that the Q[,, ]-ratonal onts n D (H) wth deth at most 5 are, n horoshercal coordnates: Deth : (, ) and ts Γ -translates; Deth : (, + ) and ts Γ -translates; Deth 4: (, ), (, + + ), ( +, ++ ) and ther Γ -translates; Deth 5: (, 4+ 5 ) and ts Γ -translates. Integral lfts of these onts are: