A Magma package to recognise finitely generated discrete subgroups of SL(2, R) and PSL(2, R).

Types

GrpSL2RGen

A finite generating set for a subgroup of SL(2, R). R is represented by an algebraic number field together with a place (real embedding). This type has the following attributes:

• field - The base field.
• matalg - The matrix algebra.
• place - The place (real embedding) of the field.
• seq- Sequence of generators of the group.

The following attributes are initially unset, and are set by RecognizeDiscreteTorsionFree(gen::GrpSL2RGen).

• has_neg - True if psl is false and the group contains -I.
• neg_word - -I as a word in the generators.
• type - The type of group. This attribute is one of the following:
  • "un" - Unknown type (only used for intermediate reduction steps).
  • "df" - Discrete and free.
  • "dc" - Discrete with co-compact action.
  • "ab" - Contains an indiscrete abelian subgroup.
  • "el" - Contains an elliptic element.
• witness - The witness proving that the group is a particular type. For "un", this is a Nielsen-equivalent generating set. For "df" and "dc", this is a reduced generating set (not including -I). For "sm" and "ab", this is a pair of elements generating an indiscrete group. For "el", this is an elliptic element.
• witness_word - A word corresponding to each element of the witness.

The following attributes are only defined if SL2Gen is determined to be discrete and torsion-free.

• FPgrp - A finitely-presented group isomorphic to this group.
• matgrp - A matrix group (with reduced generating set) isomorphic to this group.
• iso - Isometry from matgrp to FPgrp.

Intrinsics

Discrete and free groups

SL2Gens(seq::SeqEnum[AlgMatElt[FldNum]], place::PlcNumElt : psl := false) -> GrpSL2RGen

Create a generating set for a subgroup of SL(2, R). If psl is set, then the generators will be considered to be representatives of elements of PSL(2, R).

RecognizeDiscreteTorsionFree(gen::GrpSL2RGen)

Decide a generating set of SL(2, R) is discrete and torsion-free. This adds data to gen that records the results of the recognition algorithm.

IsDiscreteTorsionFree(gen::GrpSL2RGen) -> BoolElt

Return true if the generating set is discrete and torsion-free.

IsDiscreteFree(gen::GrpSL2RGen) -> BoolElt

Return true if the generating set is discrete and free.

IsDiscreteCocompact(gen::GrpSL2RGen) -> BoolElt

Return true if the generating set is discrete with cocompact action.

IsElementOf(g::AlgMatElt, gen::GrpSL2RGen) -> BoolElt, GrpFPElt

Decide whether g is an element of the group, returning the word in the reduced set evaluating to g.

MapToFundamentalDomain(z::Tup, gen::GrpSL2RGen) -> AlgMatElt, GrpFPElt

Return g (and corresponding word w) such that gz is in the fundamental domain. Two points in the same orbit will be mapped to the same orbit representative.

ReducedGenerators(gen::GrpSL2RGen) -> SeqEnum[AlgMatElt]

Return a reduced generating set for a discrete torsion-free group.

BaseField(gen::GrpSL2RGen) -> FldNum

Return the base field of the group.

HasNegativeOne(gen::GrpSL2RGen) -> FldNum, GrpFPElt

Return true if the subgroup of SL(2, R) has -I.

Rank(gen::GrpSL2RGen) -> RngIntElt

The rank of a discrete torsion-free group.

Discrete groups

TorsionFreeSubgroup(gen::GrpSL2RGen) -> GrpSL2RGen, SetEnum[AlgMatElt], RngIntElt

Find a generating set for a torsion-free congruence subgroup.

IsDiscrete(gen::GrpSL2RGen) -> BoolElt, GrpSL2RGen, SetEnum[AlgMatElt], RngIntElt

Decide whether a subgroup of SL(2, R) or PSL(2, R) is discrete, returning a finite index subgroup and set of coset representatives if so.

IsElementOf(g::AlgMatElt, tf_gp::GrpSL2RGen, cosets::SetEnum[AlgMatElt]) -> BoolElt, GrpFPElt, AlgMatElt

Decide whether g is an element of a subgroup of SL(2, R) or PSL(2, R), given a torsion-free discrete subgroup and a set of coset representatives. If it is, then return (w, s) where s is a coset representative and w is a word in the reduced set such that w*s evaluates to g.