Fundamental Domains#

Let \(D\) denote the unit disc and \(\Gamma\) an arithmetic Fuchsian group. A fundamental domain in \(D\) for \(\Gamma\) is a closed, hyperbolically convex region \(P \subset D\) such that the translates of \(P\) by \(\Gamma\) cover \(D\) and such that the interiors of all translates \(P\) are disjoint, i.e. \(D=\bigcup_{\gamma \in \Gamma} \gamma P\) and \({\rm int}(P) \cap {\rm int}(\gamma P) = \emptyset\) for \(\gamma \neq 1\), where \({\rm int}(P)\) denotes the interior of \(P\).

Choosing a point \(p \in D\) not fixed by any element of \(\Gamma \setminus \{1\}\), we obtain a fundamental domain by letting

\[P=\{z \in D:d(z,p) \leq d(z,\gamma(p)){\rm\ for\ all\ }\gamma \in \Gamma\}.\]

Typically, one takes \(p=0\).

One can similarly define a fundamental domain in the upper half-plane \({\mathbb{H}}\), and one can bijectively map fundamental domains in \(D\) to those in \({\mathbb{H}}\) via a choice of conformal map between them. The unit disc is a more natural setting for our algorithms.

For each \(\gamma \in \Gamma \setminus \{1\}\), we define the isometric circle of \(\gamma\) to be the circle

\[C(\gamma)=\{z \in {\mathbb{C}}: |\gamma(z)|=|z| \}.\]

Then in fact \(P\) is the closure of the intersection of half-spaces given by

\[\bigcap_{\gamma \in \Gamma \setminus \{1\}} C(\gamma)^o\]

where \(C(\gamma)^o\) denotes the exterior of \(C(\gamma)\). Our algorithm recursively finds elements in \(\Gamma\) to decrease the hyperbolic volume of this intersection until the volume \({\rm vol}(X)\) is reached; it relies upon a “reduction theory” with respect to a set of generators of \(\Gamma\).

FundamentalDomain(G, D): GrpPSL2, SpcHyd -> SeqEnum#: Computes a fundamental domain in the unit disc \(D\) for the action of \(G\).

FundamentalDomain(G): GrpPSL2 -> SeqEnum#: Computes a fundamental domain in the upper half-plane for the action of \(G\).

Example: Fundamental Domains (ex-456f00)#

We first compute a fundamental domain for a quaternion algebra defined over \({\mathbb{Q}}(\zeta_7)^+\) which turns out to be the \((2,3,7)\)-triangle group.

> K<z> := CyclotomicField(7);
> F := sub<K | z+1/z >;
> b := F! (z+1/z);
> A<i,j,k> := QuaternionAlgebra<F | b, b>;
> O := MaximalOrder(A);
> G := FuchsianGroup(O);
> P := FundamentalDomain(G, UnitDisc());
> P;
[
    0.0563466917619454773195578124639 -
      0.265288162495691167957067899257*$.1,
    0.0563466917619454773195578124639 +
      0.265288162495691167957067899257*$.1,
    -0.0886504855947700264615254294500 -
      4.57194956512909992886313419322E-100*$.1
]
> P := FundamentalDomain(G);
> P;
[
    0.496970425395180896221180392445 +
      (0.867767478235116240951536665696)*root(-1),
    -0.496970425395180896221180392445 +
      (0.867767478235116240951536665696)*root(-1),
    6.94379666203368024633240684073E-100 +
      (0.753423227948677598725236624130)*root(-1)
]
> ArithmeticVolume(P);
0.0238095238095238095238095238092
> ArithmeticVolume(G);
1/42
> ($1)*1.0;
0.0238095238095238095238095238095

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We can visualize the domain \(P\) by using the postscript plotting tools of Chapter Congruence Subgroups of \({\operatorname{PSL}}_2({\mathbb{R}})\).

> DisplayPolygons(P, "/tmp/quat237triang.ps" : Show := true);
[ -0.496970425395180896221180392445, 0.496970425395180896221180392445,
1.05000000000000000000000000000, 302.000000000000000000000000000 ]

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We repeat this with the quaternion algebra over \({\mathbb{Q}}\) of discriminant \(10\).

> G := FuchsianGroup(QuaternionOrder(10));
> P := FundamentalDomain(G);
> DisplayPolygons(P, "/tmp/quat10.ps" : Show := true);
[ -1.31448412972780982023636168140, 0.158113883008418966599944677206,
1.27912476227312394698179550056, 235.000000000000000000000000000 ]
> U, m := Group(G);
> U;
Finitely presented group U on 3 generators
Relations
    U.1^3 = Id(U)
    U.2^3 = Id(U)
    U.3^3 = Id(U)
    (U.1^-1 * U.2^-1 * U.3^-1)^3 = Id(U)

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ShimuraReduceUnit({delta, }{gammagens, G, D}): AlgAssVOrdElt, SeqEnum[AlgAssVOrdElt], GrpPSL2, SpcHyd -> SeqEnum#

ShimuraReduceUnit(delta, gammagens, G, D): AlgQuatElt, SeqEnum[AlgQuatElt], GrpPSL2, SpcHyd -> SeqEnum#

CreateWord  : BoolElt                      Default: false
NextSmallest: BoolElt                      Default: false
z0          : SpcHydElt                    Default: 0
z1          : SpcHydElt                    Default: 0

Reduce the unit delta moving \(z_0\) to \(z_1\) with respect to the generators in \(\gamma_{{\rm gens}}\) by multiplying on the left or right by elements of \(\gamma_{{\rm gens}}\) inside the arithmetic Fuchsian group \(G\) to minimize the distance to the origin in the unit disc \(D\). Returns a sequence of triples containing reduced elements and the sequence on the left and right to obtain them. The argument CreateWord reduces \(\delta\) even if \(\delta\) is in \(\gamma_{{\rm gens}}\), disallowing the trivial word.