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the hyperbolic dehn filling theory was established by w. thurston in his original note @xcite . besides the general theory , he extensively analyzed the space of dehn fillings of the figure eight knot complement in terms of the decomposition by two ideal tetrahedra . the analysis leaded us to extremely fascinated research activities on dehn fillings in the last 25 years . one of unsolved problems in his analysis was to determine the global shape of the dehn filling space . since then , there have been many deep researches such as @xcite discussing this problem in fact , however it is still mysterious . even , though w. neumann and a. reid @xcite have determined the true shape for the whitehead link complement with one cusp component being complete , there seems to be no concrete examples of `` one - cusped '' hyperbolic manifolds or orbifolds for which we see the true shape of the dehn filling space . in this paper , we construct hopefully the first such example of an orbifold , regretfully rather than a manifold . the example we present here is topologically @xmath0-covered as an orbifold by the complement of a seven component link in the connected sum of three copies of @xmath1 . based on the study of one - circle packings on complex affine tori by the second author in @xcite , we construct the example , which we will denote by @xmath2 , and its all possible dehn filling deformations in the following three sections . then using careful analysis of deformations of such tori in @xcite again , we determine the global shape of the dehn filling space of @xmath2 in subsequent two sections . we start with the _ hexagonal packing _ @xmath3 on the complex plane @xmath4 by equi - radii circles , see figure [ fig : hexagonalpacking ] . the set of euclidean translations which leave @xmath3 invariant forms a group @xmath5 acting freely on @xmath4 . it is isomorphic to @xmath6 and the quotient is a particular euclidean torus admitting a cyclic symmetry of order @xmath7 , which we call the _ hexagonal torus_. @xmath3 then descends to what we call a one - circle packing on the hexagonal torus . ( 100,100 ) ( 0,0 )    let us regard the complex plane @xmath4 as the boundary of the upper half space model @xmath8 of the hyperbolic 3-space . @xmath3 together with the dual packing @xmath9 which consists of circumscribed circles of interstices in @xmath3 fills @xmath4 . each member of both @xmath3 and @xmath9 bounds a hemisphere in @xmath8 and such a hemisphere bounds a hemiball facing @xmath4 . cutting off those hemiballs from @xmath8 , we obtain a region @xmath10 with ideal polygonal boundary in @xmath8 . the intersections of @xmath3 and @xmath9 correspond to ideal vertices of @xmath10 with rectangular section . the group @xmath5 acts properly discontinuously on @xmath8 by the poincar extensions and the region @xmath10 is invariant under the action . hence taking quotient , we obtain a hyperbolic 3-manifold @xmath2 with ideal polygonal boundary and a cusp . the polygonal boundary consists of two ideal triangles and one ideal hexagon , where they intersect orthogonally . take the double of @xmath2 along triangular faces , and then take again the double of the result along the boundary which now consists of two ideal hexagons . this double doubling construction gives us a hyperbolic 3-manifold @xmath11 with seven cusps among which 4 are old and 3 new . it is obvious by the construction that @xmath11 admits a @xmath12-symmetry generated by two reflections associated to the doubling . @xmath2 is the quotient of @xmath11 by this symmetry and therefore we regard @xmath2 as an orbifold from now on , and study its dehn filling deformations as an orbifold . topologically @xmath2 is a solid torus with a core loop and three points on the boundary removed . to see the effect on the first double , stretch the removed vertices on @xmath13 to arcs appropriately so that they become circles after doubling . then by first doubling , we obtain a genus 3 handle body with two core loops and three loops on the boundary removed . this is depicted in figure [ fig : firstdouble ] , where the shaded disks represent triangular faces along which we took the double . ( 141,123 ) ( 0,0 )    a framed link representation of @xmath11 , the double of figure [ fig : firstdouble ] , can be immediately described by figure [ fig : framedlink ] . ( 145,110 ) ( 0,0 ) ( 45,72)@xmath14 ( 81,72)@xmath14 ( 117,72)@xmath14 we will construct dehn filling deformations of @xmath2 using deformations of the hexagonal packing on @xmath4 with the same combinatorial type . the hexagonal packing descends to a one - circle packing on the hexagonal torus . conversely , a one - circle packing on the torus combinatorially equivalent to one on the hexagonal torus is universal covered by a packing on the plane equivalent to the hexagonal packing . however since the hexagonal packing is rigid in euclidean geometry , there are no other one - circle packings on euclidean tori than one on the hexagonal torus . hence to get deformations , we will work with one - circle packings not on euclidean but rather on complex affine tori . this section is to review what has been known about complex affine tori and circle packings on them according to @xcite . a _ complex affine structure _ on the torus is by definition a collection of local charts on @xmath4 whose transition function is a restriction of a complex affine transformation , so in particular it defines a complex structure . the uniformization assigning the conformal class to each affine torus defines a map of the space @xmath15 of marked complex affine tori to the teichmller space @xmath16 of the torus . it is a natural complex line bundle projection , where the fiber is parameterized by explicit description of developing maps of affine structures under some normalization as we will see below . @xmath16 is identified with the upper half plane @xmath17 in @xmath4 and hence @xmath15 is homeomorphic to the product @xmath18 . we call the first factor a _ teichmller parameter _ @xmath19 and the second an _ affine parameter _ @xmath20 . complex affine tori sharing the same @xmath21 are conformally equivalent . we denote by @xmath22 the marked complex affine torus associated to @xmath23 . also we denote by @xmath24 specified simple closed curves generating the fundamental group of the torus for marking . the zero section of the line bundle @xmath25 precisely consists of euclidean structures . in this case , the developing map @xmath26 is a homeomorphism , and the images of @xmath27 and @xmath28 by the holonomy @xmath29 are translations defined for each @xmath30 by @xmath31 after appropriate normalization . ( 250,120 ) ( 0,20 ) ( 13,0)images of @xmath27 and @xmath28 ( 150,20 ) ( 145,0)image of a circle packing    in affine but non - euclidean case , identifying the universal cover @xmath32 with @xmath33 by the uniformization , the developing map is described by an exponential map @xmath34 under the identification of @xmath15 with @xmath18 . here @xmath21 and @xmath35 are the teichmller and affine parameters of the complex affine torus in question . the images of @xmath36 by the holonomy @xmath29 are obviously similarities @xmath37 which fix the origin . ( 250,120 ) ( 0,20 ) ( 13,0)images of @xmath27 and @xmath28 ( 150,20 ) ( 145,0)image of a circle packing    since a complex affine transformation is a similarity , it sends a circle on @xmath4 to a circle . hence we can still discuss about circle packings on complex affine tori . the second author showed in @xcite that the set of complex affine tori which admit a one - circle packing forms a real two - dimensional moduli space @xmath38 in @xmath18 . in constructing the moduli space , the marking for the packing was specified so that the edges of the nerve of the packing are homotopic to @xmath27 , @xmath28 and @xmath39 . the moduli space @xmath38 contains only one euclidean torus associated to @xmath40 , which is the hexagonal one . moreover , the restriction of the uniformization @xmath25 to @xmath38 is onto and two to one except at the hexagonal torus . the affine parameters of a generic fiber consist of nonzero @xmath35 and its negative @xmath41 . to each pair @xmath42 on the moduli space @xmath43 , we construct a dehn filling deformation of @xmath2 as an orbifold based on the construction of @xmath2 . to see this , we start with a complex affine torus @xmath44 packed by one circle combinatorially equivalent to the one - circle packing on @xmath45 . the developing map of @xmath44 is an exponential map and @xmath46-covers @xmath47 . thus if we let @xmath48 be the universal cover of @xmath49 , it canonically lifts to a homeomorphism @xmath50 . by mapping a circle packing on @xmath32 to @xmath48 , we obtain a circle packing @xmath51 on @xmath48 combinatorially equivalent to the hexagonal packing on @xmath4 . also the holonomy lifts uniquely to an injective homomorphism of @xmath52 to the group of transformations of @xmath48 . we denote its image by @xmath53 . then we let @xmath54 denote the geodesic in @xmath8 joining @xmath14 and @xmath55 . the universal cover @xmath56 of @xmath57 will be a substitution of @xmath8 in the euclidean case . in fact , cutting off hemiballs in @xmath56 bounded by the members of a deformed hexagonal packing @xmath51 and its dual @xmath58 on @xmath48 , we obtain a region @xmath59 with ideal polygonal boundary in @xmath56 . also since the action of @xmath53 on @xmath48 extends to a properly discontinuous action on @xmath56 which leaves @xmath59 invariant , by taking a quotient we obtain a hyperbolic 3-manifold with ideal polygonal boundary and incomplete end near @xmath54 . we denote its completion by @xmath60 . it carries a dehn surgery type singularity by definition , see @xcite , and we thus obtain to each @xmath42 in @xmath61 a dehn filling deformation of @xmath62 . on the other hand , let @xmath63 be a dehn filled deformation of @xmath2 as an orbifold and @xmath64 a manifold obtained from @xmath63 by removing the points attached by the completion . @xmath64 is homeomorphic to the torus times @xmath65 with three points on the boundary removed . the boundary of @xmath64 consists of two ideal triangles and one ideal hexagon , where they intersect orthogonally . choose a developing map @xmath66 of @xmath64 so that a neighborhood of an incomplete end winds @xmath54 in @xmath8 . by this normalization , the holonomy of @xmath64 maps @xmath67 to transformations fixing @xmath14 and @xmath55 , the image of the holonomy contains an element of infinite order by the definition of the dehn filling deformation . @xmath64 is not complete , but not so wild . let @xmath68 be an equidistant neighborhood of an incomplete end and @xmath69 its complement . it is obvious that any point in @xmath64 admits the unique shortest path to @xmath70 . thus @xmath68 is developed to an equidistant neighborhood of @xmath54 and @xmath69 its complement . in particular , the image of the developing map @xmath66 misses @xmath54 , and @xmath66 is in fact a map of the universal cover @xmath71 into @xmath57 . then @xmath66 lifts to a map @xmath72 of the universal cover @xmath71 to @xmath56 , the universal cover of @xmath57 . accordingly , the holonomy of @xmath64 lifts to a homomorphism to the group of transformations of @xmath56 . we denote its image by @xmath73 . let @xmath74 be the preimage of the boundary of the equidistant neighborhood of @xmath54 in @xmath56 . @xmath73 acts properly discontinuously and freely on @xmath74 by the definition of the dehn filling deformation , and @xmath74 bijectively corresponds to the preimage of @xmath74 by @xmath72 . this shows that @xmath72 is injective at least on a neighborhood of the preimage of @xmath74 by @xmath72 . we will show that @xmath72 is globally injective . suppose we have two points in @xmath71 mapped by @xmath72 to a point @xmath75 . @xmath76 admits the unique shortest path to the point @xmath77 on @xmath74 . since the preimage of @xmath74 by @xmath72 bijectively corresponds to the preimage of @xmath70 in @xmath71 , the preimage of @xmath77 consists of a single point @xmath78 . hence two points in question lie on the orthogonal path to the preimage of @xmath70 through @xmath78 on the same side with the same distance to @xmath78 . thus they must be the same . then the ideal hexagonal faces on the boundary of @xmath71 in the developed image define a packing on @xmath48 , the universal cover of @xmath47 , combinatorially equivalent to the hexagonal one . moreover the packing is invariant under the induced action of @xmath73 on @xmath48 . hence it descends to a one - circle packing on some affine torus @xmath22 parameterized by some pair @xmath79 , and @xmath63 is isometric to @xmath60 . following @xcite , we review how to find a pair @xmath42 lying on the moduli space @xmath61 . if @xmath42 lies on @xmath61 , a simple observation shows that @xmath21 and @xmath35 must satisfy the identities , @xmath80 moreover , the inequalities @xmath81 must be satisfied by obvious reason . let @xmath82 be a smooth function defined by @xmath83 where @xmath84 is the largest possible region for @xmath35 to vary in @xmath38 . the identity ( [ eq : conditionforadmittingpacking ] ) shows that @xmath35 , @xmath85 and @xmath86 lie on the same level set @xmath87 . when @xmath88 , @xmath89 consists of a single point corresponding to the hexagonal torus by simple computation . for each @xmath90 , @xmath91 is topologically a simple loop around the origin , has symmetries about the real and imaginary axis , and bounds a strictly convex interior .    for @xmath92 , the origin @xmath14 , @xmath35 , @xmath85 and @xmath86 form a parallelogram by obvious reason in plane geometry . conversely , if @xmath93 is given and let @xmath91 be the level set on which @xmath35 lies , then there are unique two points @xmath94 on @xmath91 so that @xmath95 lie in counterclockwise order on @xmath91 , and they together with the origin form a parallelogram . then we obtain a pair @xmath42 in the moduli space @xmath61 by letting @xmath96 . hence @xmath97 determines @xmath21 . in view of this fact , we identify the moduli space @xmath61 with the region @xmath98 . in the level set @xmath91 for @xmath90 , there are twelve special points depicted in figure [ fig : twelvepointsonlevelset ] which give rise to pairs on @xmath61 rather obviously . here , @xmath99 and @xmath100 are intersections with the real axis . similarly @xmath101 , @xmath102 lie on the imaginary axis . @xmath103 for @xmath104 are points such that @xmath105 , @xmath106 , @xmath107 and @xmath108 are the bisectors of the segments @xmath109 , @xmath110 , @xmath111 and @xmath112 where @xmath14 is the origin in @xmath113 . three points @xmath114 , @xmath115 , @xmath116 together with the origin @xmath14 form a parallelogram and give rise to a pair on @xmath61 . as @xmath117 decreases from @xmath118 toward @xmath14 , an ellipselike closed curve @xmath91 shrinks down to a point @xmath89 , however the order of @xmath103 s on the level set never changes . hence the trace of @xmath119 defines a locus @xmath120 in the region @xmath121 , and @xmath121 is divided by these loci into twelve regions denoted by @xmath122 , @xmath123 , @xmath124 as in figure [ fig : twelveregions ] . ( 140,70 ) ( 0,-7 ) ( 0,0 ) , title="fig : " ] ( 123,35)(0,0)[lb]@xmath99 ( 119,53)(0,0)[l]@xmath125 ( 98,63)(0,0)[lb]@xmath126 ( 77,65)(0,0)[b]@xmath101 ( 49,63)(0,0)[rb]@xmath127 ( 25,53)(0,0)[r]@xmath128 ( 20,30)(0,0)[r]@xmath100 ( 25,18)(0,0)[r]@xmath129 ( 49,7)(0,0)[rt]@xmath130 ( 70,5)(0,0)[rt]@xmath102 ( 91,7)(0,0)[lt]@xmath131 ( 119,21)(0,0)[l]@xmath132 ( 69,34)(0,0)[tr]@xmath14    ( 140,70 ) ( 0,-7 ) ( 0,0 ) , title="fig : " ] ( 144,35)(0,0)[l]@xmath133 ( 144,49)(0,0)[l]@xmath134 ( 144,63)(0,0)[l]@xmath135 ( 72,69)(0,0)[l]@xmath136 ( -4,63)(0,0)[r]@xmath137 ( -4,49)(0,0)[r]@xmath138 ( -4,35)(0,0)[r]@xmath139 ( -4,21)(0,0)[r]@xmath140 ( -4,7)(0,0)[r]@xmath141 ( 70,2)(0,0)[r]@xmath142 ( 144,7)(0,0)[l]@xmath143 ( 144,21)(0,0)[l]@xmath144 ( 105,42)(0,0)[]@xmath122 ( 98,56)(0,0)[t]@xmath145 ( 77,60)(0,0)[t]@xmath146 ( 63,60)(0,0)[t]@xmath147 ( 42,56)(0,0)[t]@xmath148 ( 35,42)(0,0)[]@xmath149 ( 35,28)(0,0)[]@xmath150 ( 42,14)(0,0)[b]@xmath151 ( 63,11)(0,0)[b]@xmath152 ( 79,11)(0,0)[b]@xmath153 ( 98,14)(0,0)[b]@xmath154 ( 105,28)(0,0)[]@xmath124    to each pair @xmath42 on the moduli space @xmath61 , we choose four points @xmath155 which form a parallelogram . choosing the origin and the third and fourth vertices @xmath156 , and adding the negative of the second one at the end , we obtain a next parallelogram spanned by @xmath157 . doing the same shift twice , we obtain parallelograms spanned by @xmath158 and @xmath159 . the last one is symmetric to the original parallelogram about the origin . this shows that the moduli space @xmath160 has a rotational symmetry of order @xmath7 . on the other hand , the complex conjugation gives an orientation reversing involutive symmetry . thus by these all together , the moduli space @xmath161 admits a dihedral symmetry of order @xmath162 . regarding @xmath36 as a meridian - longitude pair for @xmath2 , we discuss the hyperbolic dehn filling coefficients of @xmath60 . recall that @xmath163 is a similarity defined by @xmath164 and @xmath165 by @xmath166 for nonzero @xmath35 . then the dehn filling coefficients @xmath167 for @xmath60 is by definition , see @xcite , a continuous solution of the equation @xmath168 under the normalization that @xmath169 go to the identity when the structure approaches the complete one . more precisely for our choice of @xmath35 and @xmath21 , the dehn filling coefficient @xmath170 of @xmath60 is a solution of the linear equation @xmath171 note that a pair @xmath42 corresponds to @xmath172 and conversely a pair @xmath173 to @xmath174 . we define @xmath173 for @xmath175 as @xmath176 .    to each finite dehn filling coefficient @xmath170 , we assign a slopelike number @xmath177 , where the slope is @xmath178 by definition . though @xmath42 corresponds to coordinates @xmath172 , the unique @xmath179 is assigned for them . dividing ( [ eq : coefficient ] ) on both sides by @xmath180 and taking the real parts , we obtain an obvious identity , @xmath181 which shows how to read off @xmath179 with respect to the dehn filling coefficient of @xmath60 by an elementary plane geometry in @xmath4 as follows . @xmath35 and @xmath85 lie on the same level set of @xmath182 in @xmath98 . here we draw the straight line through @xmath35 and @xmath85 . then the intersection of the line with the imaginary axis is the point which divides the segment connecting @xmath35 and @xmath85 in @xmath183 to @xmath179 .    for our convenience , we define a function @xmath184 by assigning @xmath177 to each @xmath185 . it is easy to see that the level set of @xmath186 at @xmath187 coincide with the union of @xmath133 and @xmath139 , the union of @xmath134 and @xmath140 , @xmath123 , the union of @xmath138 and @xmath144 . when @xmath35 moves along the level set of @xmath182 in counterclockwise direction , @xmath85 does the same . hence @xmath186 is strictly increasing along a counterclockwise move on the level set of @xmath182 except a gap at @xmath188 . ( 140,70 ) ( 0,-15 ) ( 0,10 ) ( 60,85)@xmath189 ( 145,72)@xmath190 ( 145,58)@xmath191 ( 145,44)@xmath192 ( 145,30)@xmath193 ( 145,16)@xmath194 ( 60,0)@xmath189 ( -28,72)@xmath194 ( -32,58)@xmath193 ( -34,44)@xmath192 ( -28,30)@xmath191 ( -37,16)@xmath190    choose a point @xmath35 in the region @xmath146 , then @xmath179 assigned lies in @xmath195 . when @xmath35 moves along the level set of @xmath186 toward the boundary of @xmath121 , @xmath85 must move along the level set of @xmath186 in @xmath148 . since @xmath196 goes to @xmath197 and @xmath198 remains zero , @xmath199 must go to @xmath55 . the limit of the curve @xmath135 is @xmath200 , and this gives @xmath201 . on the other hand , by ( [ eq : conditionforadmittingpacking ] ) , @xmath202 . since @xmath203 is bounded and @xmath204 is asymptotically @xmath205 , @xmath206 converges to @xmath14 and the limit of @xmath207 is @xmath208 . thus , @xmath209 , and we have @xmath210 this shows that @xmath211 and @xmath212 .    by the dihedral symmetry of the moduli space , we can calculate the limit for other @xmath179 . the image of the dehn filling coefficients @xmath173 parameterized by the moduli space @xmath213 then covers the shaded region @xmath214 in figure [ fig : dehn surgery space ] , and the boundary consists of a hexagon as follows : @xmath215    ( 115,110 ) ( 0,-10 ) ( 0,0 ) ( 2,122)@xmath192 ( 13,120 ) ( 50,115)@xmath191 ( 105,105)@xmath190 ( 105,59)@xmath189 ( 105,25)@xmath194 ( 105,0)@xmath216 ( 105,51)@xmath217 ( 50,105)@xmath218 ( 54,82)@xmath219 ( 80,55)@xmath219 ( 37,18)@xmath220 ( 11,44)@xmath220 we check how the degeneration occurs when they approach the boundary . we first think of the deformations along the path defined by the level set of @xmath186 at @xmath189 . then the deformations are realized by hyperbolic cone - manifolds and the meridian loop @xmath27 bounds a disk hitting the cone singularity . see @xcite for details about cone - manifolds . since @xmath221 the cone angle converges to @xmath208 and the length of the singular locus diverges . in fact , the dehn filled manifold splits along an euclidean sub cone - manifold , a disk with two corner of angle @xmath222 and a cone point of cone angle @xmath208 , at the limit . the same result holds when @xmath191 and @xmath193 because of the dihedral symmetry .    more generally , choose coprime integers @xmath223 and @xmath224 and think of the deformations of @xmath2 along the path on the level set of @xmath186 at @xmath225 . then again the deformations are realized by cone - manifolds and a loop homotopic to @xmath226 bounds a disk hitting the cone singularity . for @xmath227 , the level set of @xmath186 lies in @xmath228 . when the deformations approach the boundary along this path , @xmath229 , @xmath230 , @xmath231 and @xmath232 . the calculation similar to one in the previous section shows that @xmath233 and hence the cone angle converges to @xmath234 . to see the length of the singular locus , we choose integers @xmath235 and @xmath117 such that @xmath236 . then , @xmath237 represents a curve which intersects @xmath238 once and hence the translation factor of @xmath239 , or equivalently @xmath240 represents the length of the singular locus . making use of the property @xmath241 , we have @xmath242 which implies that the length of the singular locus diverges and a corresponding cone - manifold splits as before . we can see similar facts for other rational @xmath179 by the dihedral symmetry . in fact , the cone angle approaches @xmath243 if @xmath244 and @xmath245 if @xmath246 . in either case , the length of the singular locus diverges . if we relax the complex affine structure modeled on @xmath4 to the complex projective structure modeled on the riemann sphere , the relaxation defines the involution @xmath247 on @xmath18 which sends @xmath248 to @xmath249 . the moduli space @xmath250 of complex projective tori admitting a one - circle packing is the quotient of @xmath38 by @xmath247 . this space is studied in @xcite by the cross ratio parameters and represented as a semi algebraic set by @xmath251 see the appendix in @xcite .    assigning the holonomy representation to each projective structure , we obtain a map @xmath252 of @xmath250 to the space @xmath253 of representations of the fundamental group of the torus in @xmath254 up to conjugation . it is a regular map with respect to the algebraic structures on the source and target , and easily seen to be a proper injective homeomorphism by concrete computation in @xcite . this in particular implies also that the boundary we described here is the true one since holonomy representations diverge at the boundary . 99 d. cooper , c. hodgson and s. kerckhoff : _ three - dimensional orbifolds and cone - manifolds _ , msj memoirs , * 5 * ( 2000 ) . j. dowty : _ orthlength and hyperbolic dehn surgery _ thesis , melbourne univ . , r. c. gunning : _ affine and projective structures on riemann surfaces _ , riemann surfaces and related topics : ann . of math . , * 97 * ( 1981 ) , 225 - 244 . c. hodgson : _ degeneration and regeneration of geometric structures on three - manifolds _ , ph.d . thesis , princeton univ . , 1986 . s. kojima , s. mizushima and s. p. tan : _ circle packings on surfaces with projective structures _ , preprint . available at http://www.arxiv.org/abs/math.gt/0111214 . s. mizushima : _ circle packings on complex affine tori _ , osaka j.math . , * 37 * ( 2000 ) , 873 - 881 . w. d. neumann and a. reid : _ arithmetic of hyperbolic manifolds _ , topology 90 , ed . by b. apanasov et al , de gruyter , ( 1992 ) , 273 - 310 . thurston : the geometry and topology of 3-manifolds , lecture notes , princeton univ . , 1977/78 .
we construct the first example of a `` one - cusped '' hyperbolic 3-orbifold for which we see the true shape of the space of hyperbolic dehn fillings .