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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="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0

<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

When span =""\D=overlineF \ap B}) where closurehas span "impleMath>\\ D\<> four - <=>(\/>ofjava.lang.StringIndexOutOfBoundsException: Range [371, 370) out of bounds for length 1069

It is explained in <a href="chapBib_mj.html#biBswanB">[Swa71b]</a> that \(F\cap B\) is a \(3\)-cell in the above mentioned regular CW-complex structure on \(X\).

<a id="X7B9DE54F7ECB7E44" name="X7B9DE54F7ECB7E44"></a>

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

Explicit fundamental domains for certain values of More recently a GAP implementation of Swan's algorithm has been included in HAP. The implementation java.lang.StringIndexOutOfBoundsException: Index 169 out of bounds for length 0

<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C">involving of minimum radius 1/9java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

<h4>14.4 The fundamental domainjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 83

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

<div java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [81, 5) out of bounds for length 95
3-dimensional
p em>usp vertex of of =">\\)/s> : 2 they byred dots.Computer calculations show that these cusps lie in precisely three orbits under the action of \(G_{-d}\). Thus, together with the orbit of \(\infty\) 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 \(\mathbb Q(\sqrt{-39})\) is of order \(4\).
and non-cuspidal vertices of minimum squared height 1/49 .

gap<pan class="APprompt"gap&
gap&java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0

/>/>

<img

A

The following additional commands comvert the Bianchi polyhedron \(D\) to a regular CW-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>

<panclass"APprompt">gap&;/ span ">:=D;<java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
Regular CW-complex of dimension 2

<img src="images/bianchi3D22.png" align="center" height="550" alt="Fundamental domain for a Bianchi group"/> <img src=

</pre></div>

<img src="images/bianchi1skeleton.gif" align="center" java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 0

A span class="GAP<><class"OQ,D<pre>
="java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 96
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 86
gap> Display2D(OQ,D);;<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</pre>h4>145 spanclass"

Two cusps are visible in

A<pan class"APprompt>gap></pan>P:=BianchiPolyhedron(-46);

<div class="example"><pre>
gap> Display3D(OQ,D);;
<an class="GAPinput">Display2D(OQ,D);;/span>

</pre></divSwanBianchiCriterion);

<img src="images/bianchi3D163.png" align="center" height="550" alt="Fundamental domain for a Bianchi group"/> <img src=t testif theof waslarge.If testreturns thenyou\

<> is a orbit ofcuspsin this , orbit containing \(\infty\), since \(\mathbb Q(\sqrt{-163})\)</ich canbe edited inthe hap//bianchig .

A fundamental domain for \(G_{-33}\) is visualizeddimensionalBianchi overOQ (

<div class="example"><pre
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 96
java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
<>a="78476F127B73BBD1"name"78476F127B73BBD1"<a/>

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

<a id=java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0

<h4>14.5 :=BianchiGcomplex(-43);;

<java.lang.StringIndexOutOfBoundsException: Range [6, 2) out of bounds for length 611

For a few values of \ >apg;/ span =GAPinputC(R;<spanjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95

<div class="example"><pre>
gap> P:=BianchiPolyhedron(-46);
Try
 :=BianchiPolyhedronO,N)
for java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 SwanBianchiCriterion(P);
to test if the value of N was large enough. If the test returns false then you\
'll need to try a larger value of N.

A successful valueap&;span>spanclass"APinput">[.0],-HomologyR,);/
ichcan edited manually the file /libCongruence/ianchi. .

<span class=" [ 2, 4, 12 ],,4,12],[ 2 ,2 ,[2,2 2, 12 ,[2 2 ,24 ,
3-dimensional Bianchi polyhedron over OQ( Sqrt(
-))involving of squared 1/
441 and non-cuspidal vertices of minimum squared

gap> SwanBianchiCriterion(P);
true

<pre></iv>

<a id[[0],[6 0, 0,0, 0 0 ] [2,2, 2, ,12 0, 0,0,0 ]

<h4>14.6 Computing a free resolution for \(SL_2({\mathcal

The above fundamental domains can be

<div class="example"><pre>
gap></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> commands the of orbits cusps i addition to the orbitofspan class"SimpleMath">\(\infty\)). They determine that there is precisely one
&;/><spanclass">([1.K.dimension()]-&tOrder(K!.stabilizer(0,k)));
gap> List([0..10],n->Homology(C,n));
[ [ 0 ], [ 0, 0 ], [ 2, 2, 12, 0 ], [ 2, 2, 24 ], [ 2, 2 ],
 [ 2, <classjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 128

gap> R:=FreeGResolution(K,11);;
gap> ist0.,gHomologyTensorWithIntegers)));gap> <
gap> R:=FreeGResolution(K,11);;
gap> a id"X7E5A36D47F9D4A47" name="X7E5A36D47F9D4A47"></a>
[ [0], [ 6 , 0,0,0,0] [2,2, 2 4,12 0,0,0,0 ]
 [ 2, 2, 2, 2, 24 ], [ 2, 2, 2, 4, 12 ], [ 2, 2, 2, 2, 24 ],
 [ 2,2, 2,4,12 ], [2, 2,, 2 2, 24] ,4 12 ],
 [ 2, 2, 2, 2, 24 ], java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</pre></div>

/ spanclass=GAPinputchi=;<spanjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77

<div class="example"><pre>
<<spanclass="GAPprompt">> for k in [1..K!.dimension(n)] do
gap> g:=Order(K!.stabilizer(n,k));
, , ,4 4 ,12 ]

gap> K:=BianchiGcomplex(-10);;
apgt;/span> Order(K!.stabilizer(0,k)));
[ 6, 6, 4, 4, 6, infinity ]

gap> gap> List([1..K!.dimension(0)],k->Order(K!.stabilizer(0,k)));
[ 6, 6, 2, 2, 2, infinity, infinity, 2, infinity, 6, 4 ]

</pre></div>

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

<h4>14.7 Some sanity checks</h4>

There is ample scope for bugs in the implementation of the above method for computing resolutions of Bianchi groups. The following sanity checks lend confidence to the implementation.

<a id="X7E5A36D47F9D4A47" name="X7E5A36D47F9D4A47"></a>

<h5>14.7-1 Equivariant Euler characteristic</h5>

Let One easy test to make in our computations is to check that the equivariant Euler characteristic of the \(2\)-complex is indeed java.lang.StringIndexOutOfBoundsException: Index 164 out of bounds for length 0

<div class="example"><pre>
/>< =GAPinput">:0;/span>
gap> K:=BianchiGcomplex(-23);;
fork [.K.(n)do/>
> g:=Order(K!.stabilizer(n,k));
<class=GAPprompt>gt;/span><class"APinput"ifg lt thenchi= +-)n/;;/>
> od;od;
<">ap&; <pan class="APinputchi
0

</pre></div> [ ],[ ,[] [] ] [] ,[] ] ] ] ]

<a id< =">&t</> <pan =:=;List([.R.dimension(n)]k-&t;ResolutionBoundaryOfWord(R,n-1,R!.boundary(n,k)));

<h5>14.7-2 Boundary , [ ], [ ] [ ] [ ] [ ],java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74

The signs in the ] ] ]]

<div class="example"><pre>
gap> gap>ap&;< class=GAPinput"n:2;([.Rdimension),-gtR(R,n,R!.(nk));span>
[ [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ],
 [ ], [ ], [ ], [ ], [ ], [ ], [ ], [[ ],[ ],[ ], [ ] [ ] [ ], [ ] [ ] [ ] [ ] [ ] [ ],
 ],[ ] [ ] ]
gap> n:=3;;List([1..R!.dimension(n)],k->ResolutionBoundaryOfWord(R,n-1,R!.boundary(n,k)));
[ [ ], [ ], [ ], </pre></iv
 [ ], [ ], [ ], [ ], <><aid="7E64819A7C058EDD" nameX7"></a</>
 [ ], [ ], [ ] ]
gap> java.lang.StringIndexOutOfBoundsException: Range [0, 63) out of bounds for length 0
[[ ], [ ] [] ]] ] ] [] [ ,[] [ ] ]
 [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [
 [ , [ ], ]]

</pre></div>

>

<h5>14.7-3 Compare different span class="GAPprompt">gap> List([0..9],n->Homology(TensorWithIntegers(R),n));< , ,,00] [, ,2 ,12, ,2 ,java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74

Sebastian Schoennenbeck in his thesis work computed some contractible \(2\)-complexes on which Bianchi groups act java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 0

The following commands test span class="GAPprompt">gap>< G>ist[.9]n&;omologyTensorWithIntegers()n);

<div , , ,212, , ,[, ]
gap> K
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
14.7span "eading"Compareto algebra/>/>
[ [ 0
 [ 2, 2, 12 ], [ 2, 2, 12 ], [ 2, 2, 12 ], [ 2, 2, 12 ], [ 2, 2, 12 ] ]

gap> K:=BianchiGcomplex(-23);;
pre
[ 0, 0 ], [2 2,12] 2 2 12 ,
 [ 2, 2,spanclass"APprompt>gt;/ span class=GAPinput"!d(0]k&tO(!stabilizer,))

</pre></div>

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

<h5>14.7-4 Compare geometry to algebra</h5>

<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 502

<div class="example"><pre>
gap> K:=BianchiGcomplex(-23);;
</div>

A visualization of the fundamental domain tells us a certain amount about the algebra. In the case of \(SL_2({\mathcal O}_{-23})\)

<img src="images/sl2O-23.png" align=" x*k^-2*r^-1*x*r^-1*s^-*k^-1*s^-,x^-1**k^3*s*r*x^-1*s*r ]

a span classjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 101

<div class="example"><>gapgt</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 77
< class"GAPprompt"gapgt;gap> List([1..K
[, 4 ]

</pre></div>

<a id="X78BC9D077956089A" name="X78BC9D077956089A

<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> gapgt;/span spanclass">:R!.elts[22]</pan
 
gap> P:=PresentationOfResolution(R);
gap> G:=SimplifiedFpGroupspan class""ap&;<spanclass=GAPinput">w:=R!.elts[2];
<fp group on the generators [ k, r, s, w, x ]>
gap> RelatorsOfFpGroup(G);
[ w3/2 ++ 1/ Sqrt(-23, -3/ +12 Sqrt(-23)],
 ^1*w*x^-*sr*w^-1**s^-1, ^1k^-**^-,
 x*k^-2*r^-1*x*r^-1*s^-1*k^-1*s^-1, x^-1*k^3*s*r*x^-1*s*r ]

gap> #Next we identify the generators as matrices
spanclass=GAPprompt">ap&t;/pan>GeneratorsOfGroup(P!.freeGroup);
[ k, m, n, p, q, r, s, t, u, v, w, x, y, z ]
<panclass"APprompt"gap> P!.gens;
[ 19, 6, 6, 20, 6, 21, 22, 6, 52, 53, 2, 50, 1, 4 ]

<pan class=GAPprompt"gap>/span=!elts[19]
[ [ 1, 1 ],
 [ -1, 0 ] ]
span "GAPinput"r:R.[21;/span
[ [ 3, 3 + -1 Sqrt(-23) ],
 [ -3/2 + -1/2 Sqrt(-23), -5 ] ]
gap> s:=R!.elts[22];
[ [ 2 + 1 Sqrt(-23), 13/2 +
 52 + 1/2 Sqrt(-23), -1 Sqrt(-23) ] ]
gap> w:=R!.elts[2];
[ [ 3/2 + 1/2 Sqrt(-23), -3/2 + 1/2 Sqrt(-23) ],
 [ 3/2 + -1/2 Sqrt(-23), 3 ] ]
<java.lang.StringIndexOutOfBoundsException: Range [30, 5) out of bounds for length 84
[ [ 11/2 + 1/2 Sqrt(-23), 15/2 + -1/2 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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