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<a id="X805848868005D528" name="X805848868005D528"></a>
<div class="ChapSects"><a href="chap14_mj.html#X805848868005D528">14 Fundamental domains for Bianchi groups</a>
<div class="ContSect"> <a href="chap14_mj.html#X858B1B5D8506FE81">14.1 Bianchi groups</a>

</div>
<div class="ContSect"> <a href="chap14_mj.html#X872D22507F797001">14.2 Swan's description of a fundamental domain</a>

</div>
<div class="ContSect"> <a href="chap14_mj.html#X7B9DE54F7ECB7E44">14.3 Computing a fundamental domain</a>

</div>
<div class="ContSect"> <a href="chap14_mj.html#X7A489A5D79DA9E5C">14.4 Examples</a>

</div>
<div class="ContSect"> <a href="chap14_mj.html#X86CD59CB7A04EE5A">14.5 Establishing correctness of a fundamental domain</a>

</div>
<div class="ContSect"> <a href="chap14_mj.html#X78476F127B73BBD1">14.6 Computing a free resolution for \(SL_2({\mathcal O}_{-d})\)</a>

</div>
<div class="ContSect"> <a href="chap14_mj.html#X784B2156823AEB15">14.7 Some sanity checks</a>

<div class="ContSSBlock">
 <a href="chap14_mj.html#X7E5A36D47F9D4A47">14.7-1 Equivariant Euler characteristic</a>

 <a href="chap14_mj.html#X852CDAFF84C5DF01">14.7-2 Boundary squares to zero</a>

 <a href="chap14_mj.html#X7E64819A7C058EDD">14.7-3 Compare different algorithms or implementations</a>

 <a href="chap14_mj.html#X8223864085412705">14.7-4 Compare geometry to algebra</a>

</div></div>
<div class="ContSect"> <a href="chap14_mj.html#X78BC9D077956089A">14.8 Group presentations</a>

</div>
<div class="ContSect"> <a href="chap14_mj.html#X786CFAA17C0A6E7A">14.9 Finite index subgroups</a>

</div>
</div>

<h3>14 Fundamental domains for Bianchi groups</h3>

<a id="X858B1B5D8506FE81" name="X858B1B5D8506FE81"></a>

<h4>14.1 Bianchi groups</h4>

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 \(\mathbb Q(\sqrt{-d})\). 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 \(\partial {\frak h}^3\) to be the Riemann sphere \(\mathbb C \cup \infty\) and let \(\overline{\frak h}^3\) denote the union of \({\frak h}^3\) and its boundary. The action of \(G_{-d}\) extends to the boundary. The element \(\infty\) and each element of the number field \(\mathbb Q(\sqrt{-d})\) are thought of as lying in the boundary \(\partial {\frak h}^3\) and are referred to as cusps. Let \(X\) denote the union of \({\frak h}^3\) with the set of cusps, \(X={\frak h}^3 \cup \{\infty\} \cup \mathbb Q(\sqrt{-d})\). 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 <a href="chapBib_mj.html#biBswanB">[Swa71b]</a>.

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 \(\le 2\).

<a id="X872D22507F797001" name="X872D22507F797001"></a>

<h4>14.2 Swan's description of a fundamental domain</h4>

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 \(a\ne 0\). A unimodular pair can be represented by a hemisphere in \(\overline{\frak h}^3\) with base centred at the point \(b/a \in \mathbb C\) and of radius \(|1/a|\). The radius is \(\le 1\). Think of the points in \({\frak h}^3\) as lying strictly above \(\mathbb C\). Let \(B\) denote the space obtained by removing all such hemispheres from \({\frak h}^3\).

When \(d \equiv 3 {\rm \ mod\ } 4\) let It is explained in <a href="chapBib_mj.html#biBswanB">[Swa71b]</a> that <a id="X7B9DE54F7ECB7E44" name="X7B9DE54F7ECB7E44"></a>

<h4>14.3 Computing a fundamental domain</h4>

Explicit fundamental domains for certain values of \(d\) were calculated by Bianchi in the 1890s and further calculations were made by Swan in 1971 <a href="chapBib_mj.html#biBswanB">[Swa71b]</a>. In the 1970s, building on Swan's work, <a href="https://www.sciencedirect.com/science/article/pii/S0723086913000042">Robert Riley</a> developed a computer program for computing fundamental domains of certain Kleinian groups (including Bianchi groups). In their 2010 PhD theses <a href="https://theses.hal.science/tel-00526976/en/">Alexander Rahm</a> and <a href="https://wrap.warwick.ac.uk/id/eprint/35128/">M.T. Aranes</a> independently developed Pari/GP and Sage software based on Swan's ideas. In 2011 <a href="https://mathstats.uncg.edu/sites/yasaki/publications/bianchipolytope.pdf">Dan Yasaki</a> used a different approach based on Voronoi's theory of perfect forms in his Magma software for fundamental domains of Bianchi groups. <a href="http://www.normalesup.org/~page/Recherche/Logiciels/logiciels-en.html">Aurel Page</a> developed software for fundamental domains of Kleinian

More recently a GAP implementation of Swan's algorithm has been included in HAP. The implementation uses exact computations in \(\mathbb Q(\sqrt{-d})\) and in \(\mathbb Q(\sqrt{d})\). 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.

<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>

<h4>14.4 Examples</h4>

<ndamental domain <class="impleMath"\=overline{ \ }<>(heretheoverline denotes ) boundary(2\-cells containedinthe four vertical quadrilateral span class=SimpleMath"\2\)cells \(\partial F\). 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 A fundamental domain for <pre>
gap> D:=BianchiPolyhedron(-39);
3-dimensional Bianchi polyhedron over OQ( Sqrt(-39) )
involving hemispheresminimum squared radius 1/
and non-cuspidal vertices of minimum squared height 1/49 .

gap> gap> Display2D(D);;

</pre></div>

<img src="images/bianchi3D39.png" align="center" height="550" alt="Fundamentaln(-39);

<>A<>usp <class>(\</pan> isanyvertex of <spanclass"impleMath"\D lying in \(\mathbb C \cup \infty\). In the above visualizations for \(G_{-39}\) several cusp vertices in \(\mathbb C\)</pan are :inthe2-dimensionalvisualizationtheyarerepresented java.lang.StringIndexOutOfBoundsException: Range [421, 420) out of bounds for length 899

The following additional commands comvert the Bianchi polyhedron \(D\) to a regular CW-complex and then display its \(1\)-skeleton.

<div class="example"><pre>
gap>
<pan "APprompt>gt;/> < class=GAPinput"YRegularCWComplex);
Regular CW-complex of dimension 2

div

<<

<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

A fundamental domain for \(G_{-22}\) can be visualized using the following commands.

<"><
&t<spanclass="APinput">Y=egularCWComplex();
gap> Display3D(OQ,D);;
</div>

<java.lang.StringIndexOutOfBoundsException: Range [11, 2) out of bounds for length 223

Two cusps are visible in the visualizations for \(G_{-22}\). They lie in a single orbitjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

A fundamental domain for \(G_{-163}\) can be visualized using the following commands.

<div class="example"><pre>
gap> D:=BianchiPolyhedron(-163);;
gap> Display3D(OQ,D);;
gap> Display2D(OQ,D);;

</pre></div>

<img src="images/bianchi3D163.png" align="center" height="550" alt="Fundamental domain for a Bianchi group"/> <img src="images/bianchi2D163.png" align="center" width="350" alt="Fundamental domain for a Bianchi group"/>

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

A fundamental domain for \(G_{-33}\) is visualized using the following commands.

<div class="example"><pre>
gap> gap>
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0

</<spanclass="GAPprompt>gap> D:=BianchiPolyhedron(-22);;

<img src="images/bianchi3D33.png" align="center" height="550" alt=<panclass="GAPprompt">gap> Display3D(OQ,D);;

<a id="X86CD59CB7A04EE5A" name="X86CD59CB7A04EE5A"></a>

<>14..5 Establishing correctness of a fundamental domain</h4>

Forjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div
<=G"gapgts>java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 95
Try
 Bjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 29
for some guessed positive integer value gap> <sp;<span
 (Pjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
o if value Nwas enough the false you\
'll need to try a larger value of N.

ApThereisjust single inthis example thejava.lang.StringIndexOutOfBoundsException: Range [68, 67) out of bounds for length 254
ich can be manually file/libCongruence.i

gap> P:=BianchiPolyhedron(-46,600);
3- Bianchi polyhedron (Sqrt
-46) ) involving hemispheres of minimum squared radius 1/
441 and non-cuspidal vertices of minimum squared height 1/8280 .

gap> D:=BianchiPolyhedron(-33);;
true

</pre></div>

<< idX78476 ="78476>/>/>

<h4>epre>div

The above fundamental

<div class="example"><pre>
 The cusps of a fundamental domain can be calculated independently of the domain computation
<spanclass="GAPprompt">&t<class"GAPinput">:=TensorWithIntegers);/span>
gap>
[ [ 0 ], [ 0, 0 ]=java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 95
 P(QN;

gap> gap> apgt;/>< =G">ist(0..0],-gt;Homology(TensorWithIntegers()k)
[ [ 0 ], [ 2, 2, 0, 0, be edited manually inthefile hap/Congruence/ianchigi.
 [ 2 4,12 ,[ 2,2 ,6] 2 ,, 12 ] [,2,2 24]
 [ 2, 4, 12 ], [ 46 involvinghemispheresminimum radius/


 ,0 0 ,0,, ,, ,4 ,0 , ,
 [ 2, 2, 2, 2, 24 ], [ 2, 2, 2, 4, 12 ], [ 2, 2, 2, 2, 24 ],

 [ 2, 2, 2, 2, 24 ], [ 2, 2, 2, 4, 12 ] ]

</pre></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

Thefollowing count number ofcusps(n additionto orbit of <span=java.lang.StringIndexOutOfBoundsException: Range [118, 114) out of bounds for length 547

<div class="example"><pre>
gap> K:=BianchiGcomplex(-43);;
< class="GAPprompt"gapgtgap> K:=BianchiGcomplex(-10);;
span ="GAPprompt">gap> List([1..K!.dimension(0)],k->Order(K!.stabilizer(0,k)));
[6, 6 4, 4, 6, infinity java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

gap>">ist([..0],k-&t;(TensorWithIntegers(R,k)
</div>

<a id="X784B2156823AEB15" name="X784B2156823AEB15"></a>

<h4>14.7 Some sanity checks</h4>

class"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 95

<> =java.lang.StringIndexOutOfBoundsException: Range [33, 27) out of bounds for length 62

<h5>14.7-1 One easy test to make in our computations is to check that , , ,[,2 ,2 ][ 2, 2, 2 4,12]

<div class="example"><pre>
gap> >span>< "">:0;/>
gap> for
java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 98
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 95
> if g < infinity then [24 24,6,6 4, 4,12, 12
&;span ="java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 128
gap> chi;
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

</pre></div>

<a id="X852CDAFF84C5DF01" name="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>14.7-2 Boundary squares to zero</h5>

java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<div class="<panclass=GAPinput>chi:;
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 94
gap> R:=FreeGResolution(K,10)< class="APprompt">gt<pan=GAPinput>for in1.!dimension) 
gap> span class="&span class="> &;infinity chi:chi+(-)/;fi</spanjava.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 114
[ [ ], [ ], APprompt>gt< < =G">;/spanjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
 [ ] ] ,[ ,[ ,[ ,[ ] ,[ ,[ ,[ ,[ ,
 [ ], [ ], [
<panclass"GAPprompt"gapg;/pan <class"GAPinput">n=;ist1.!dimension]&java.lang.StringIndexOutOfBoundsException: Range [127, 124) out of bounds for length 157
,]][] ]
 [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ],
 [ ,[ ,[
<span java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
[ [ ], [ ], [ ]
 [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ],
 [ , [ ], []]
gap> n:=5;;List([1..R!.dimensionspan class=G">gt/ span class="=;List1.!.(n]k&;esolutionBoundaryOfWord-1R.oundary,k);/java.lang.StringIndexOutOfBoundsException: Index 157 out of bounds for length 157
 , , , , ,[] ] ,[, ,[, ,
 [ ], [ ] , ,[ java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 [ ], [ ], [ ] ]

/>

p "E64819A7C058EDD="7E64819A7C058EDD>/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

< , ] ,[] [] ] ] [ ,[ ,[] ,[ ]

Sebastian Schoennenbeck in his thesis work computed some contractible ,[ ] ]]

The following commands test that Sebastian Schoennenbeck's <span""R=(,0);/>
<java.lang.StringIndexOutOfBoundsException: Range [60, 5) out of bounds for length 122
[[0],[12 0 0, 0 ] [2, 2,12, 0, 0 ], [ 2, 2, 12 ] [2,2,12 ]
 [ 2, 2, 12 ], [ 2, 2, 12 ], [ 2, 2, 12 ], [ 2, 2, 12 ], [ 2, 2, 12 ] ]

gap> K:=BianchiGcomplex(-23);;
gap> R:=FreeGResolution
 <spanclass="APinput">ist(0.9,-gtH(TensorWithIntegersR,)</>
[ [ 0 ], [ 12, 0, 0, 0 ], [ 2, 2
 [ 2,2, 12],[ 2 2,, 12 ] [2,2,12 ] [2, 2, 12 ], [ 2,2,12 ]

</pre></div>

<a id="X8223864085412705" name="X8223864085412705"></a>

<-4 geometry to algebra<span<h5>

The number of cusps (i.e. the number of orbits of vertices with infinite stabilizer subgroup) must be precisely one less than the number of elements in the ideal class java.lang.StringIndexOutOfBoundsException: Range [0, 176) out of bounds for length 0

<div="<>
gap></[ [ 0 ], [ 12, 0, 0, 0 ], [ 2, 2, 12[,2 ,[2,2,12]
< =G"gap>< class="List([1..K.imension0),-g;rderK.(0k);/
[ 6, 2, 2, 4, infinity, infinity ]

</pre></div>

A visualization of the fundamental domain tells us a certain amount java.lang.StringIndexOutOfBoundsException: Range [0, 76) out of bounds for length 0

<

a fundamental domain forp>The number of cusps (i.e. the number of orbits of vertices with infinite stabilizer subgroup)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<6, 2 2,,infinity,infinity]
gap> K:=BianchiGcomplex(-23);;
gap> List([1..K!.dimension(2)],k->Length(K!.boundary(2,k)));
[ 10, 4, 4, 4, 4, 4 ]

</pre></div>

<a id="X78BC9D077956089A" name="X78BC9D077956089A"></a>

<h4>14.8 Group presentations</h4>

Swan's reason for studying fundamental domains was to obtain explicit group presentations for \(SL_2({\mathcal O}_{-d})\) for various values of \(d\). The following commands obtain a presentation for \(SL_2({\mathcal O}_{-23})\).

<div class="example"><pre>
gap> K:=BianchiGcomplex(-23);;
gap> R:=FreeGResolution(K,2);;
gap> P:=PresentationOfResolution(R);
gap> G:=SimplifiedFpGroup(P!.freeGroup/P!.relators);
<fp group on the generators [ k, r, s, w, x ]>
gap> RelatorsOfFpGroup(G);
java.lang.String: green'>2 Sqrt(-23) ],
 [ -1 Sqrt(-23), -4 + -1 Sqrt(-23) ] ]

</pre></div>

<a id="X786CFAA17C0A6E7A" name="X786CFAA17C0A6E7A"></a>

<h4>14.9 Finite index gapgt;span class">:BianchiGcomplex-);<>

The following commands compute the span class="GAPprompt">gap> <span="GAPinput">R:=QuadraticToCyclotomicCoefficients(R);;

<div class="example"><pre>
gap> OQ:=RingOfIntegers(QuadraticNumberField(-23));;
gap> I:=QuadraticIdeal(OQ,[Sqrt(-23)]);
ideal of norm 23 in O(Q(Sqrt(-23)))
gap> G:=HAP_CongruenceSubgroupGamma0(I);
<group of 2x2 matrices in characteristic 0>
gap> IndexInSL2O(G);
24

gap> K:=BianchiGcomplex(-23);;
gap> R:=FreeGResolution(K,11);;
gap> R:=QuadraticToCyclotomicCoefficients(R);;
gap> S:=ResolutionFiniteSubgroup(R,G);;
gap> List([0..10],n->Homology(TensorWithIntegers(S),n));
[ [ 0 ], [ 2, 0, 0, 0, 0, 0, 0, 0, 0 ],
 [ 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],
 [ 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],
 [ 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],
 [ 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2, 2, 2 ],
 [ 2, 2, 2, 2, 2, 2, 2, 2 ] ]

gap> P:=PresentationOfResolution(S);;
gap> H:=SimplifiedFpGroup(P!.freeGroup/P!.relators);
<fp group on the generators [ f8, f10, f15, f70, f86, f125, f132, f138, f182,
 f187, f191, f273, f279 ]>
gap> Length(RelatorsOfFpGroup(H));
24

</pre></div>

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