Construction of Congruence Homomorphisms

In this section, \(K\) is a finite degree extension of \(F(x_1, \ldots, x_m)\), where \(F\) is \(Q\), a number field, or a finite field. Also \(m \geq 0\) if char \(F\) = 0, and \(m > 0\) otherwise.

CongruenceImage(G : parameters): GrpMat -> GrpMat, HomGrp, []

Virtual  : BoolElt                      Default: false
Prime    : RngIntElt                    Default: 3
Limit    : RngIntElt                    Default: 10
ExtDegree: RngIntElt                    Default: 1

If \(G\) is a finitely generated subgroup of \({\operatorname{GL}}(n, K)\), then \(G\) has a normal subgroup \(N\) whose torsion elements are unipotent; so \(N\) is torsion-free if \(K\) has characteristic 0.

This function constructs a congruence homomorphism from \(G\) into \({\operatorname{GL}}(n, {\bf F}_{q})\) for some prime power \(q\); its kernel is \(N\). If char \(K\) is positive, then \({\bf F}_{q}\) has the same characteristic.

For a detailed description of the congruence homomorphisms see [Detinko et al., 2013, Section 3]. The function returns the congruence image \(H\), the congruence homomorphism, and the list of images of generators of \(G\).

If the optional parameter Virtual is set to true then the congruence homomorphism satisfies additional properties [Detinko et al., 2011]. In particular it can be used to test whether \(G\) satisfies the “virtual” properties described in Section Deciding Virtual Properties of Linear Groups.

The optional parameter Prime applies if \(K\) has characteristic 0: if Prime is positive, then it is a lower bound for the characteristic of the congruence image; if it is 0 then the function returns a congruence image defined over a field of characteristic 0.

The optional parameter Limit applies to groups defined over (rational) function fields. If char \(K > 0\), then we consider extensions of \(F\) to degree Limit only; otherwise we examine tuples in the ring of integers mod Limit.

The optional parameter ExtDegree applies to groups defined over (algebraic) function fields of positive characteristic: we construct a congruence image over an extension of (at least) this degree of coefficient field.