  
  [1X14 [33X[0;0YFundamental domains for Bianchi groups[133X[101X
  
  
  [1X14.1 [33X[0;0YBianchi groups[133X[101X
  
  [33X[0;0YThe  [13XBianchi  groups[113X are the groups [22XG_-d=PSL_2(cal O_-d)[122X where [22Xd[122X is a square
  free  positive integer and [22Xcal O_-d[122X is the ring of integers of the imaginary
  quadratic field [22XQ(sqrt-d)[122X. These groups act on [13Xupper-half space[113X[133X
  
  
  [24X[33X[0;6Y{\frak h}^3 =\{(z,t) \in \mathbb C\times \mathbb R\ |\ t > 0\}[133X
  
  [124X
  
  [33X[0;0Yby the formula[133X
  
  
  [24X[33X[0;6Y\left(\begin{array}{ll}a&b\\  c  &d  \end{array}\right)\cdot  (z+tj)  \  = \
  \left(a(z+tj)+b\right)\left(c(z+tj)+d\right)^{-1}\[133X
  
  [124X
  
  [33X[0;0Ywhere  we  use the symbol [22Xj[122X satisfying [22Xj^2=-1[122X, [22Xij=-ji[122X and write [22Xz+tj[122X instead
  of [22X(z,t)[122X. Alternatively, the action is given by[133X
  
  
  [24X[33X[0;6Y\left(\begin{array}{ll}a&b\\  c  &d  \end{array}\right)\cdot  (z+tj)  \  = \
  \frac{(az+b)\overline{(cz+d)  }  + a\overline c t^2}{|cz +d|^2 + |c|^2t^2} \
  +\ \frac{t}{|cz+d|^2+|c|^2t^2}\, j \ .[133X
  
  [124X
  
  [33X[0;0YWe  take  the  boundary  [22X∂  frak  h^3[122X to be the Riemann sphere [22XC ∪ ∞[122X and let
  [22Xoverlinefrak  h^3[122X  denote the union of [22Xfrak h^3[122X and its boundary. The action
  of  [22XG_-d[122X  extends  to  the  boundary.  The element [22X∞[122X and each element of the
  number  field  [22XQ(sqrt-d)[122X  are thought of as lying in the boundary [22X∂ frak h^3[122X
  and  are  referred  to as [13Xcusps[113X. Let [22XX[122X denote the union of [22Xfrak h^3[122X with the
  set  of cusps, [22XX=frak h^3 ∪ {∞} ∪ Q(sqrt-d)[122X. It follows from work of Bianchi
  and  Humbert  that  the space [22XX[122X admits the structure of a regular CW-complex
  (depending  on  [22Xd[122X)  for  which  the  action of [22XG_-d[122X on [22Xfrak h^3[122X extends to a
  cellular  action on [22XX[122X which permutes cells. Moreover, [22XG_-d[122X acts transitively
  on  the [22X3[122X-cells of [22XX[122X and each [22X3[122X-cell has trivial stabilizer in [22XG_-d[122X. Details
  are provided in Richard Swan's paper [Swa71b].[133X
  
  [33X[0;0YWe  refer  to  the closure in [22XX[122X of any one of these [22X3[122X-cells as a [13Xfundamental
  domain[113X  for  the  action  [22XG_-d[122X.  Cohomology  of  [22XG_-d[122X can be computed from a
  knowledge of the combinatorial structure of this fundamental domain together
  with a knowledge of the stabilizer groups of the cells of dimension [22X≤ 2[122X.[133X
  
  
  [1X14.2 [33X[0;0YSwan's description of a fundamental domain[133X[101X
  
  [33X[0;0YA  pair  [22X(a,b)[122X of elements in [22Xcal O_-d[122X is said to be [13Xunimodular[113X if the ideal
  generated by [22Xa,b[122X is the whole ring [22Xcal O_-d[122X and [22Xane 0[122X. A unimodular pair can
  be  represented by a hemisphere in [22Xoverlinefrak h^3[122X with base centred at the
  point  [22Xb/a  ∈ C[122X and of radius [22X|a|[122X. The radius is [22X≤ 1[122X. Think of the points in
  [22Xfrak  h^3[122X  as  lying  strictly  above  [22XC[122X. Let [22XB[122X denote the space obtained by
  removing all such hemispheres from [22Xfrak h^3[122X.[133X
  
  [33X[0;0YWhen [22Xd ≡ 3 mod 4[122X let [22XF[122X be the subspace of [22Xoverlinefrak h^3[122X consisting of the
  points  [22Xx+iy+jt[122X with [22X-1/2 ≤ x ≤ 1/2[122X, [22X-1/4 ≤ y ≤ 1/4[122X, [22Xt ≥ 0[122X. Otherwise, let [22XF[122X
  be  the  subspace  of [22Xoverlinefrak h^3[122X consisting of the points [22Xx+iy+jt[122X with
  [22X-1/2 ≤ x ≤ 1/2[122X, [22X-1/2 ≤ y ≤ 1/2[122X, [22Xt ≥ 0[122X.[133X
  
  [33X[0;0YIt  is  explained  in  [Swa71b] that [22XF∩ B[122X is a [22X3[122X-cell in the above mentioned
  regular CW-complex structure on [22XX[122X.[133X
  
  
  [1X14.3 [33X[0;0YComputing a fundamental domain[133X[101X
  
  [33X[0;0YExplicit  fundamental  domains  for  certain  values of [22Xd[122X were calculated by
  Bianchi  in  the  1890s  and  further calculations were made by Swan in 1971
  [Swa71b].   In   the   1970s,   building   on   Swan's  work,  Robert  Riley
  ([7Xhttps://www.sciencedirect.com/science/article/pii/S0723086913000042[107X)
  developed  a  computer  program for computing fundamental domains of certain
  Kleinian  groups  (including  Bianchi  groups).  In  their  2010  PhD theses
  Alexander Rahm ([7Xhttps://theses.hal.science/tel-00526976/en/[107X) and M.T. Aranes
  ([7Xhttps://wrap.warwick.ac.uk/id/eprint/35128/[107X)     independently    developed
  Pari/GP  and  Sage  software  based  on  Swan's  ideas.  In  2011 Dan Yasaki
  ([7Xhttps://mathstats.uncg.edu/sites/yasaki/publications/bianchipolytope.pdf[107X)
  used  a different approach based on Voronoi's theory of perfect forms in his
  Magma  software  for  fundamental  domains  of  Bianchi  groups.  Aurel Page
  ([7Xhttp://www.normalesup.org/~page/Recherche/Logiciels/logiciels-en.html[107X)
  developed  software  for  fundamental domains of Kleinian groups in his 2010
  masters       thesis.       In       2018       Sebastian      Schoennenbeck
  ([7Xhttps://github.com/schoennenbeck/VMH-DivisionAlgebras[107X)  used a more general
  approach  based  on  perfect  forms  in  his  Magma  software  for computing
  fundamental  domains  of  Bianchi  and other groups. Output from the code of
  Alexander  Rahm  and  Sebastian Schoennenbeck for certain Bianchi groups has
  been stored iin [12XHAP[112X for use in constructing free resolutions.[133X
  
  [33X[0;0YMore  recently a [12XGAP[112X implementation of Swan's algorithm has been included in
  [12XHAP[112X.  The  implementation  uses  exact  computations  in  [22XQ(sqrt-d)[122X  and  in
  [22XQ(sqrtd)[122X.  A  bespoke  implementation  of  these  two  fields is part of the
  implementation  so  as  to  avoid making apparently slower computations with
  cyclotomic  numbers.  The  account  of  Swan's  algorithm  in  the thesis of
  Alexander Rahm was the main reference during the implementation.[133X
  
  
  [1X14.4 [33X[0;0YExamples[133X[101X
  
  [33X[0;0YThe  fundamental domain [22XD=overlineF ∩ B[122X (where the overline denotes closure)
  has boundary [22X∂ D[122X involving the four vertical quadrilateral [22X2[122X-cells contained
  in  the four vertical quadrilateral [22X2[122X-cells of [22X∂ F[122X. We refer to these as the
  [13Xvertical  [22X2[122X-cells[113X of [22XD[122X. When visualizing [22XD[122X we ignore the [22X3[122X-cell and the four
  vertical  [22X2[122X-cells  entirely  and visualize only the remaining [22X2[122X-cells. These
  [22X2[122X-cells  can  be viewed as a [22X2[122X-dimensional image by projecting them onto the
  complex plane, or they can be viewed as an interactive [22X3[122X-dimensional image.[133X
  
  [33X[0;0YA  fundamental  domain  for  [22XG_-39[122X  can  be  visualized  using the following
  commands.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=BianchiPolyhedron(-39);[127X[104X
    [4X[28X3-dimensional Bianchi polyhedron over OQ( Sqrt(-39) ) [128X[104X
    [4X[28Xinvolving hemispheres of minimum squared radius 1/39 [128X[104X
    [4X[28Xand non-cuspidal vertices of minimum squared height 10/12493 . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XDisplay3D(D);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay2D(D);;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  [13Xcusp  vertex[113X  of  [22XD[122X  is  any  vertex  of  [22XD[122X  lying in [22XC ∪ ∞[122X. In the above
  visualizations   for  [22XG_-39[122X  several  cusp  vertices  in  [22XC[122X  are  :  in  the
  2-dimensional  visualization  they  are  represented  by  red dots. Computer
  calculations  show  that these cusps lie in precisely three orbits under the
  action  of  [22XG_-d[122X. Thus, together with the orbit of [22X∞[122X there are four distinct
  orbits  of  cusps.  By the well-known correspondence between cusp orbits and
  elements of the class group it follows that the class group of [22XQ(sqrt-39)[122X is
  of order [22X4[122X.[133X
  
  [33X[0;0YA  fundamental  domain  for  [22XG_-22[122X  can  be  visualized  using the following
  commands.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=BianchiPolyhedron(-22);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay3D(OQ,D);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay2D(OQ,D);;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTwo  cusps are visible in the visualizations for [22XG_-22[122X. They lie in a single
  orbit. Thus, together with the orbit of [22X∞[122X, there are two orbits of cusps for
  this group.[133X
  
  [33X[0;0YA  fundamental  domain  for  [22XG_-163[122X  can  be  visualized using the following
  commands.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=BianchiPolyhedron(-163);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay3D(OQ,D);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay2D(OQ,D);;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere  is just a single orbit of cusps in this example, the orbit containing
  [22X∞[122X, since [22XQ(sqrt-163)[122X is a principle ideal domain and hence has trivial class
  group.[133X
  
  [33X[0;0YA fundamental domain for [22XG_-33[122X is visualized using the following commands.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=BianchiPolyhedron(-33);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay3D(OQ,D);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay2D(OQ,D);;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
