14 Fundamental domains for Bianchi groups
  
  
  14.1 Bianchi groups
  
  The  Bianchi  groups are the groups G_-d=PSL_2(cal O_-d) where d is a square
  free  positive integer and cal O_-d is the ring of integers of the imaginary
  quadratic field Q(sqrt-d). These groups act on upper-half space
  
  
  {\frak h}^3 =\{(z,t) \in \mathbb C\times \mathbb R\ |\ t > 0\}
  
  
  
  by the formula
  
  
  \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}\
  
  
  
  where  we  use the symbol j satisfying j^2=-1, ij=-ji and write z+tj instead
  of (z,t). Alternatively, the action is given by
  
  
  \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 \ .
  
  
  
  We  take  the  boundary  ∂  frak  h^3 to be the Riemann sphere C ∪ ∞ and let
  overlinefrak  h^3  denote the union of frak h^3 and its boundary. The action
  of  G_-d  extends  to  the  boundary.  The element ∞ and each element of the
  number  field  Q(sqrt-d)  are thought of as lying in the boundary ∂ frak h^3
  and  are  referred  to as cusps. Let X denote the union of frak h^3 with the
  set  of cusps, X=frak h^3 ∪ {∞} ∪ Q(sqrt-d). It follows from work of Bianchi
  and  Humbert  that  the space X admits the structure of a regular CW-complex
  (depending  on  d)  for  which  the  action of G_-d on frak h^3 extends to a
  cellular  action on X which permutes cells. Moreover, G_-d acts transitively
  on  the 3-cells of X and each 3-cell has trivial stabilizer in G_-d. Details
  are provided in Richard Swan's paper [Swa71b].
  
  We  refer  to  the closure in X of any one of these 3-cells as a fundamental
  domain  for  the  action  G_-d.  Cohomology  of  G_-d 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 ≤ 2.
  
  
  14.2 Swan's description of a fundamental domain
  
  A  pair  (a,b) of elements in cal O_-d is said to be unimodular if the ideal
  generated by a,b is the whole ring cal O_-d and ane 0. A unimodular pair can
  be  represented by a hemisphere in overlinefrak h^3 with base centred at the
  point  b/a  ∈ C and of radius |a|. The radius is ≤ 1. Think of the points in
  frak  h^3  as  lying  strictly  above  C. Let B denote the space obtained by
  removing all such hemispheres from frak h^3.
  
  When d ≡ 3 mod 4 let F be the subspace of overlinefrak h^3 consisting of the
  points  x+iy+jt with -1/2 ≤ x ≤ 1/2, -1/4 ≤ y ≤ 1/4, t ≥ 0. Otherwise, let F
  be  the  subspace  of overlinefrak h^3 consisting of the points x+iy+jt with
  -1/2 ≤ x ≤ 1/2, -1/2 ≤ y ≤ 1/2, t ≥ 0.
  
  It  is  explained  in  [Swa71b] that F∩ B is a 3-cell in the above mentioned
  regular CW-complex structure on X.
  
  
  14.3 Computing a fundamental domain
  
  Explicit  fundamental  domains  for  certain  values of d 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
  (https://www.sciencedirect.com/science/article/pii/S0723086913000042)
  developed  a  computer  program for computing fundamental domains of certain
  Kleinian  groups  (including  Bianchi  groups).  In  their  2010  PhD theses
  Alexander Rahm (https://theses.hal.science/tel-00526976/en/) and M.T. Aranes
  (https://wrap.warwick.ac.uk/id/eprint/35128/)     independently    developed
  Pari/GP  and  Sage  software  based  on  Swan's  ideas.  In  2011 Dan Yasaki
  (https://mathstats.uncg.edu/sites/yasaki/publications/bianchipolytope.pdf)
  used  a different approach based on Voronoi's theory of perfect forms in his
  Magma  software  for  fundamental  domains  of  Bianchi  groups.  Aurel Page
  (http://www.normalesup.org/~page/Recherche/Logiciels/logiciels-en.html)
  developed  software  for  fundamental domains of Kleinian groups in his 2010
  masters       thesis.       In       2018       Sebastian      Schoennenbeck
  (https://github.com/schoennenbeck/VMH-DivisionAlgebras)  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 HAP for use in constructing free resolutions.
  
  More  recently a GAP implementation of Swan's algorithm has been included in
  HAP.  The  implementation  uses  exact  computations  in  Q(sqrt-d)  and  in
  Q(sqrtd).  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.
  
  
  14.4 Examples
  
  The  fundamental domain D=overlineF ∩ B (where the overline denotes closure)
  has boundary ∂ D involving the four vertical quadrilateral 2-cells contained
  in  the four vertical quadrilateral 2-cells of ∂ F. We refer to these as the
  vertical  2-cells of D. When visualizing D we ignore the 3-cell and the four
  vertical  2-cells  entirely  and visualize only the remaining 2-cells. These
  2-cells  can  be viewed as a 2-dimensional image by projecting them onto the
  complex plane, or they can be viewed as an interactive 3-dimensional image.
  
  A  fundamental  domain  for  G_-39  can  be  visualized  using the following
  commands.
  
    Example  
    gap> D:=BianchiPolyhedron(-39);
    3-dimensional Bianchi polyhedron over OQ( Sqrt(-39) ) 
    involving hemispheres of minimum squared radius 1/39 
    and non-cuspidal vertices of minimum squared height 10/12493 . 
    
    gap> Display3D(D);;
    gap> Display2D(D);;
    
  
  
  A  cusp  vertex  of  D  is  any  vertex  of  D  lying in C ∪ ∞. In the above
  visualizations   for  G_-39  several  cusp  vertices  in  C  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  G_-d. Thus, together with the orbit of ∞ 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 Q(sqrt-39) is
  of order 4.
  
  A  fundamental  domain  for  G_-22  can  be  visualized  using the following
  commands.
  
    Example  
    gap> D:=BianchiPolyhedron(-22);;
    gap> Display3D(OQ,D);;
    gap> Display2D(OQ,D);;
    
  
  
  Two  cusps are visible in the visualizations for G_-22. They lie in a single
  orbit. Thus, together with the orbit of ∞, there are two orbits of cusps for
  this group.
  
  A  fundamental  domain  for  G_-163  can  be  visualized using the following
  commands.
  
    Example  
    gap> D:=BianchiPolyhedron(-163);;
    gap> Display3D(OQ,D);;
    gap> Display2D(OQ,D);;
    
  
  
  There  is just a single orbit of cusps in this example, the orbit containing
  ∞, since Q(sqrt-163) is a principle ideal domain and hence has trivial class
  group.
  
  A fundamental domain for G_-33 is visualized using the following commands.
  
    Example  
    gap> D:=BianchiPolyhedron(-33);;
    gap> Display3D(OQ,D);;
    gap> Display2D(OQ,D);;