Constructing a representation of the group (2, 3, 7; 11)

D.F. Holt  W. Plesken  B. Souvignier

In this note we show, how a representation of the group G := (2, 3, 7; 11) defined by the presentation ⟨x, y|x^{2}, y^{3}, (xy)^{7}, [x, y]^{11}⟩ can be constructed by the method described in [PlS 94]. Infiniteness of this group was left open in [HoP 92], but M. Edjvet proved that it is infinite by topological methods (cf. [Edj 92]). We obtain a homomorphism from G into a Lie group of type G_{2} over an algebraic number field of degree 10 over I Q, which gives infinitely many finite factor groups of G.

It can be shown using the resultant method described in Section 2 of [PlS 94] that G has no projective representation of degree 2 in characteristic 0 and one calculates that p = 43 is the only prime such that G maps onto P SL_{2}(p). It was already pointed out in [HoP 92] that P SL_{2}(43) and J_{1} are the only simple epimorphic images of G of order less than 10^{11}.

Denote by K the kernel of the epimorphism from G onto P SL_{2}(43). For any group K and prime number p define the lower p-central series λ^{i} = λ_{p}^{i} of K by λ^{1}(K) := K, λ^{i+1}(K) := [λ^{i}(K), K]λ^{i}(K)^{p}. The lower 43-central series of K starts with layers |K/λ^{2}(K)| = 43^{11}, |λ^{i}(K)/λ^{i+1}(K)| = 43^{14} for i = 2, 3, 4, where [λ^{i}(K), K] = λ^{i+1}(K) for i = 1, 2, 3, 4 and λ^{i}(K)^{43} = λ^{i+2}(K) for i = 1, 2, 3. This indicates that the 43-adic completion of K might be a subgroup of a Lie group of type G_{2} over a ramified extension E of I Q_{43} of degree 2. Under this hypothesis it follows from [Sat 71] that a representation Δ of G can be realized of degree 7 over E. Moreover E = I Q_{43}[π] with π = \sqrt{43} or \sqrt{−43} are the only ramified quadratic extensions of I Q_{43}.

The simple IF_{43}P SL_{2}(43)-module of dimension 7 gives us a homomorphism from G into SL_{7}(43). This can be lifted to SL_{7}(R/℘^{3}) for R = ZZ_{43}[\sqrt{−43}] but not for the other ramified extension, hence we set π = \sqrt{−43} and ℘ = πR. We chose ξ ∈ SL_{7}(R) as image of x and let φ_{d} : G → SL_{7}(R/℘^{d}) such that xφ_{d} is represented by ξ. As in Section 4 of [PlS 94] let L(φ_{d}) = L^{1}(φ_{d}) be the affine space over IF_{43} of lifts of φ_{d} to SL_{7}(R/℘^{d+1}) and L^{2}(φ_{d}) the subspace of those which lift to SL_{7}(R/℘^{d+2}). The centralizer C of ξ and C_{0} of ξ and yφ_{d} modulo ℘ yield dimIF_{43}C/C_{0} = 3^{2} + 4^{2} − 1, since ξ has 3 eigenvalues 1 and 4 eigenvalues -1 and Gφ_{1} is absolutely irreducible over IF_{43}. For d = 2, 3, … we find dimL^{1}(φ_{d}) = 25 and dimL^{2}(φ_{d}) = 24. As pointed out in Lemma 4.1 of [PlS 94] lifting directly from R/℘^{d} to R/℘^{d+2} is still a linear problem over IF_{43}, and this gives a unique 1-step-lift up to conjugation. To obtain a solution one has to choose values for the free parameters in each step. Clearly this choice can (and in this example did) cause that the representation is not written over its minimal splitting field. For that purpose one has to find a suitable new basis. Now one observes that the image of xy has eigenvalue 1 with multiplicity 1, and spinning an eigenvector v to this eigenvalue yields the basis B := (v, vΔ(x), vΔ(y), vΔ(yx), vΔ(yxy), vΔ(yxyx), vΔ(yxyy)). The fact that these vectors are linearly independent can already be concluded in the first step of the lift. With respect to this basis one obtains:

Δ(x) = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ a & a & b & b & c & c & -1 \end{matrix} ) and Δ(y) = ( \begin{matrix}0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ d & -d-e & e & f & g & 1 & -f-g \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} ).

Applying the method from Section 5 in [PlS 94] to a lift to R/℘^{80} one can now identify the entries with elements in the field F generated by τ := tr(Δ(xy[x, y^{−1}]^{2}xy^{−1})), which is an extension of degree 2 of I Q[ζ_{11} + ζ_{11}^{−1}], where ζ_{11} is an 11-th root of unity. Define θ := ζ_{11} + ζ_{11}^{−1}, then the minimal polynomial for τ over I Q[θ] is t^{2} + (θ^{4} + 2θ^{3} − 3θ^{2} − 5θ)t − θ^{4} − 2θ^{3} + 2θ^{2} + 7θ + 4, the minimal polynomial over I Q is t^{10} − 5t^{9} + 13t^{8} − 22t^{7} + 53t^{6} − 70t^{5} − 21t^{4} − 176t^{3} − 91t^{2} + 54t + 23. With respect to τ and θ the entries of the matrices can now be expressed as follows: a = 1/4609((−604θ^{4} + 3292θ^{3} + 3555θ^{2} − 10680θ − 1447)τ + (−735θ^{4} − 7837θ^{3} − 1481θ^{2} + 25005θ + 4443)), b = 1/4609((−3638θ^{4} − 2301θ^{3} + 11996θ^{2} + 3556θ − 3725)τ + (6485θ^{4} + 5342θ^{3} − 21328θ^{2} − 11148θ + 4067)), c = 1/4609((−669θ^{4} + 655θ^{3} + 2358θ^{2} − 2825θ + 2144)τ + (796θ^{4} + 881θ^{3} − 3281θ^{2} + 950θ + 625)), d = 1/43((−9θ^{4} − 47θ^{3} − 43θ^{2} + 70θ + 87)τ + (55θ^{4} + 34θ^{3} − 258θ^{2} − 165θ + 142)), e = 1/43(−29θ^{4} + 11θ^{3} + 215θ^{2} + 87θ − 164), f = 1/43(−21θ^{4} + 48θ^{3} + 215θ^{2} + 20θ − 141), g = 1/43((8θ^{4} + 37θ^{3} + 43θ^{2} − 24θ − 63)τ + (−4θ^{4} − 40θ^{3} − 43θ^{2} + 55θ + 53)). One might suspect that it would have been easier to work out the representation Δ as homomorphism into O_{7}(R) rather than SL_{7}(R). It is indeed so that the computation does not get more difficult: From the IF_{43}P SL_{2}(43)-module of dimension 7 one gets an invariant bilinear form over IF_{43} unique up to scalar multiples. By [Sch 85] it can be lifted to a form Φ over R unique up to isometry and also ξ can be chosen to fix Φ. The affine spaces L^{1}(φ_{d}) and L^{2}(φ_{d}) have to be adjusted to consist of orthogonal lifts and the centralizers C and C_{0} to contain only Φ-skew symmetric elements. The IF_{43}-dimensions of the adjusted C/C_{0}, L^{1}(φ_{d}) and L^{2}(φ_{d}) are 9, 10 and 9 respectively. Hence one does not gain anything. Anyhow the Gram matrix of the invariant form Φ of Δ as given over the global field F = I Q[τ] above is

( \begin{matrix}7 & f_{1} & f_{1} & f_{1} & f_{1} & f_{2} & f_{2} \\ f_{1} & 7 & f_{1} & f_{1} & f_{2} & f_{1} & f_{1} \\ f_{1} & f_{1} & 7 & f_{2} & f_{1} & f_{3} & f_{1} \\ f_{1} & f_{1} & f_{2} & 7 & f_{3} & f_{1} & f_{3} \\ f_{1} & f_{2} & f_{1} & f_{3} & 7 & f_{4} & f_{3} \\ f_{2} & f_{1} & f_{3} & f_{1} & f_{4} & 7 & f_{3} \\ f_{2} & f_{1} & f_{1} & f_{3} & f_{3} & f_{3} & 7 \end{matrix} ), where \begin{matrix}f_{1} = −2θ^{3} − 2θ^{2} + 6θ + 5, \\ f_{2} = −2θ^{4} − 4θ^{3} + 4θ^{2} + 10θ + 1, \\ f_{3} = −2θ^{4} − 2θ^{3} + 6θ^{2} + 6θ + 1, \\ f_{4} = −2θ^{4} − 2θ^{3} + 6θ^{2} + 6θ + 2τ − 1. \end{matrix}

This proves that Δ(G) lies in a non-split form of B_{3}(F), since the image of Φ under each of the real embeddings of F is positive definite. Furthermore there exists an F Ghomomorphism from Λ^{2}V onto V, where V is the module for Δ, hence the image lies in a non-split form of G_{2}(F). Using KANT (cf. [Poh 93]) we computed that the field F has discriminant 11^{8} · 43 · 353, classnumber 1, 4 real and 3 pairs of complex embeddings into I C and Galois group C_{2} ≀ C_{5}.

The last fact proves that the matrix Δ(xy[x, y^{−1}]^{2}xy^{−1}) has infinite order. There exists a representation over the ring R of integral elements in F, which can be obtained by conjugation with a suitable matrix, since on the one hand F has classnumber 1 and on the other hand this is possible for the 70-dimensional representation over I Q obtained from Δ by substituting τ by its companion matrix of degree 10. Reducing the representation modulo a maximal ideal ℘ of R with residue class degree f = 1 above the rational prime p gives epimorphisms for instance onto the following finite groups: J_{1} for p = 11, G_{2}(23) for p = 23, P SL_{2}(43) and G_{2}(43) for p = 43, G_{2}(67) for p = 67, G_{2}(89) for p = 89. Note, in the case p = 11 there are two maximal ideals above p, which together give an epimorphism onto J_{1} × J_{1}. We take the opportunity to correct a mistake in [HoP 92], where it was stated that the abelianized kernel of the epimorphism from G onto one of the J_{1}’s is a 2-group. From the results above it follows that it maps as well onto an elementary abelian group of order 11^{14}. Whether or not the 2-part of the abelianization can be used to prove that the representation Δ constructed in this paper is not faithful has not been pursued further.

Bibliography

[Edj 92] M. Edjvet, An example of an infinite group. in: Discrete Groups and Geometry (ed. by W.J. Harvey, C. MacLachlan), London Math. Soc. Lecture Notes Series 173, pp. 66-74, Cambridge University Press, 1992

[HoP 92] D.F. Holt, W. Plesken, A Cohomological Criterion For A Finitely Presented Group To Be Infinite. Journal London Math. Soc. (2), 45, pp. 469-480, 1992

[Poh 93] M. Pohst e.a, KANT-V2. Computer Algebra in Deutschland, edited by Fachgruppe Computeralgebra der GI, DMV und GAMM, pp. 212-218, 1993

[PlS 94] W. Plesken, B. Souvignier, Analyzing finitely presented groups by constructing representations. These proceedings

[Sat 71] I. Satake, Classification Theory of Semi-Simple Algebraic Groups. Lecture Notes in Pure and Applied Mathematics, New York, 1971

[Sch 85] W. Scharlau, Quadratic and Hermitian Forms. Springer, Berlin, 1985

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