<Chapter><Heading>Fundamental domains for Bianchi groups</Heading>
<Section><Heading>Bianchi groups</Heading>

The <E>Bianchi groups</E> are the groups 
<M>G_{-d}=PSL_2({\cal O}_{-d})</M> where <M>d</M> is a square free positive 
integer and <M>{\cal O}_{-d}</M> is the ring of integers
of the imaginary quadratic field <M>\mathbb Q(\sqrt{-d})</M>. 
These groups act on <E>upper-half space</E>

<Display>{\frak h}^3 =\{(z,t) \in \mathbb C\times \mathbb R\ |\ t &gt; 0\}  </Display>

	by the formula
	<Display>\left(\begin{array}{ll}a&amp;b\\ c &amp;d \end{array}\right)\cdot (z+tj) \ = \ \left(a(z+tj)+b\right)\left(c(z+tj)+d\right)^{-1}\ </Display>

where we use the symbol <M>j</M> satisfying <M>j^2=-1</M>, <M>ij=-ji</M> and write <M>z+tj</M> instead of <M>(z,t)</M>. 
Alternatively,   the action is given by
<Display>\left(\begin{array}{ll}a&amp;b\\ c &amp;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
      \ .</Display>

<P/>We take the boundary <M>\partial {\frak h}^3</M> to be the Riemann 
	sphere <M>\mathbb C \cup \infty</M> and let 
	<M>\overline{\frak h}^3</M> denote the union of <M>{\frak h}^3</M> 
and its boundary. The action of <M>G_{-d}</M> extends to the 
	boundary. The element <M>\infty</M> and each element of the number 
		field 
	<M>\mathbb Q(\sqrt{-d})</M> are thought of as lying in the boundary <M>\partial {\frak h}^3</M> and are  referred to as <E>cusps</E>. 
Let <M>X</M> denote the union of <M>{\frak h}^3</M> with the set of cusps, <M>X={\frak h}^3 \cup \{\infty\} \cup \mathbb Q(\sqrt{-d})</M>.
	It follows from work of Bianchi and Humbert that the space <M>X</M>
 admits the structure 
of a regular CW-complex (depending on <M>d</M>) for which the 
action of <M>G_{-d}</M> on <M>{\frak h}^3</M> extends to a cellular 
	action on <M>X</M> which permutes cells. Moreover, 
<M>G_{-d}</M> acts transitively on the <M>3</M>-cells 
	of <M>X</M>
and each <M>3</M>-cell has trivial stabilizer in <M>G_{-d}</M>. Details are provided in Richard Swan's paper <Cite Key="swanB"/>.

<P/> We refer to the closure in <M>X</M>
	of any one of these <M>3</M>-cells as a <E>fundamental domain</E> for the action <M>G_{-d}</M>. Cohomology of <M>G_{-d}</M> 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 <M>\le 2</M>.

</Section>

<Section><Heading>Swan's description of a fundamental domain</Heading>
		A pair <M>(a,b)</M> of elements in <M>{\cal O}_{-d}</M> is said to be <E>unimodular</E> if the ideal generated by <M>a,b</M> is the whole ring <M>{\cal O}_{-d}</M> and  <M>a\ne 0</M>.
			A unimodular pair can be represented by a hemisphere in <M>\overline{\frak h}^3</M> with base centred at the point <M>b/a \in \mathbb C</M> and of radius <M>|a|</M>. 
				The  radius is <M>\le 1</M>. 
					Think of the points in <M>{\frak h}^3</M> as lying strictly above <M>\mathbb C</M>.  Let <M>B</M>
denote the space obtained by removing all such hemispheres from <M>{\frak h}^3</M>. 

<P/> When <M>d \equiv 3 \mod 4</M> let <M>F</M> be the subspace of <M>\overline{\frak h}^3</M> consisting of the points <M>x+iy+jt</M> with <M>-1/2 \le x \le 1/2</M>, <M>-1/4 \le y \le 1/4</M>, <M>t \ge 0</M>. Otherwise, let <M>F</M> be the subspace of <M>\overline{\frak h}^3</M> consisting of the points <M>x+iy+jt</M> with <M>-1/2 \le x \le 1/2</M>, <M>-1/2 \le y \le 1/2</M>, <M>t \ge 0</M>.

<P/> It is explained in <Cite Key="swanB"/> that <M>F\cap B</M> is a <M>3</M>-cell in the above mentioned regular CW-complex structure on <M>X</M>. 
</Section>

<Section><Heading>Computing a fundamental domain</Heading>

	Explicit fundamental domains for certain
	values of <M>d</M> were calculated by Bianchi in the 1890s and further calculations were made by  Swan in 1971 <Cite Key="swanB"/>. In the 1970s, 
		building on Swan's work, <URL><Link>https://www.sciencedirect.com/science/article/pii/S0723086913000042</Link><LinkText>Robert Riley</LinkText></URL> developed a computer 
		program for computing fundamental domains of certain Kleinian groups (including Bianchi groups). In their 2010 PhD theses  <URL><Link>https://theses.hal.science/tel-00526976/en/</Link><LinkText>Alexander Rahm</LinkText></URL> and <URL><Link>https://wrap.warwick.ac.uk/id/eprint/35128/</Link><LinkText>M.T. Aranes</LinkText></URL>  
independently developed  Pari/GP and Sage software
			based on Swan's ideas. In 2011 <URL><Link>https://mathstats.uncg.edu/sites/yasaki/publications/bianchipolytope.pdf</Link><LinkText>Dan Yasaki</LinkText></URL> used a  different 
approach  based on Voronoi's theory of perfect forms in his  
		Magma software  for fundamental domains of Bianchi groups. 
			<URL><Link>http://www.normalesup.org/~page/Recherche/Logiciels/logiciels-en.html</Link><LinkText>Aurel Page</LinkText></URL> developed software for fundamental domains of Kleinian groups in his 2010 masters thesis.
				In 2018 <URL><Link>https://github.com/schoennenbeck/VMH-DivisionAlgebras</Link><LinkText>Sebastian Schoennenbeck</LinkText></URL> 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 <B>HAP</B> for use in constructing free resolutions.
		
		<P/>More recently a <B>GAP</B> implementation of Swan's algorithm has been included  in <B>HAP</B>. The implementation uses exact computations in <M>\mathbb Q(\sqrt{-d})</M> and in <M>\mathbb Q(\sqrt{d})</M>. 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.  

</Section>

<Section><Heading>Examples</Heading>
	The fundamental domain <M>D=\overline{F \cap B}</M> (where the overline denotes closure) has boundary 
	<M>\partial D</M> involving the four vertical quadrilateral
	<M>2</M>-cells  contained in the four vertical quadrilateral <M>2</M>-cells of <M>\partial F</M>. We refer to these as the <E>vertical <M>2</M>-cells</E> of <M>D</M>. 
		When visualizing <M>D</M> we ignore the <M>3</M>-cell and the four vertical <M>2</M>-cells entirely  and
		visualize only the remaining  <M>2</M>-cells.  
			These <M>2</M>-cells can be viewed as a <M>2</M>-dimensional image by projecting them onto the complex plane, or they can be viewed as an interactive <M>3</M>-dimensional image.

<P/>A fundamental domain for <M>G_{-39}</M> can
				be visualized using the following commands.

			<Example>
<#Include SYSTEM "tutex/bianchi.1.txt">
</Example>
			<Alt Only="HTML">&lt;img src="images/bianchi3D39.png" align="center" height="550" alt="Fundamental domain for a Bianchi group"/>
</Alt>
<Alt Only="HTML">&lt;img src="images/bianchi2D39.png" align="center" width="350" alt="Fundamental domain for a Bianchi group"/>
</Alt>
<P/> A <E>cusp vertex</E> of <M>D</M> is any vertex of <M>D</M>  lying
	in <M>\mathbb C \cup \infty</M>. In the above visualizations for 
	<M>G_{-39}</M> several cusp vertices in <M>\mathbb C</M>
		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 <M>G_{-d}</M>. Thus, together with the orbit of <M>\infty</M> 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 <M>\mathbb Q(\sqrt{-39})</M> is of order <M>4</M>. 
			

<P/>A fundamental domain for <M>G_{-22}</M> can
                                be visualized using the following commands.

                        <Example>
<#Include SYSTEM "tutex/bianchi.2.txt">
</Example>
                        <Alt Only="HTML">&lt;img src="images/bianchi3D22.png" align="center" height="550" alt="Fundamental domain for a Bianchi group"/>
</Alt>
<Alt Only="HTML">&lt;img src="images/bianchi2D22.png" align="center" width="350" alt="Fundamental domain for a Bianchi group"/>
</Alt>

<P/>Two cusps are  visible in the visualizations for <M>G_{-22}</M>. They 
	 lie in a single orbit. Thus, together with the orbit of <M>\infty</M>, there are two orbits of cusps for this group.


<P/>A fundamental domain for <M>G_{-163}</M> can
                                be visualized using the following commands.

                        <Example>
<#Include SYSTEM "tutex/bianchi.3.txt">
</Example>
                        <Alt Only="HTML">&lt;img src="images/bianchi3D163.png" align="center" height="550" alt="Fundamental domain for a Bianchi group"/>
</Alt>
<Alt Only="HTML">&lt;img src="images/bianchi2D163.png" align="center" width="350" alt="Fundamental domain for a Bianchi group"/>
</Alt>

<P/>There is just a single orbit of cusps in this example, the orbit containing <M>\infty</M>, since <M>\mathbb Q(\sqrt{-163})</M> is a principle ideal domain and hence has trivial class group. 

	<P/>A fundamental domain for <M>G_{-33}</M> is 
                                 visualized using the following commands.

                        <Example>
<#Include SYSTEM "tutex/bianchi.4.txt">
</Example>
                        <Alt Only="HTML">&lt;img src="images/bianchi3D33.png" align="center" height="550" alt="Fundamental domain for a Bianchi group"/>
</Alt>
<Alt Only="HTML">&lt;img src="images/bianchi2D33.png" align="center" width="350" alt="Fundamental domain for a Bianchi group"/>
</Alt>


</Section>
</Chapter>
