Hyperbolic unit groups and quaternion algebras. (English) Zbl 1186.16044
The authors classify the quadratic extensions \(K=\mathbb{Q}[\sqrt d]\) (with \(d\) an integer) and the finite groups \(G\) so that the group \(\mathcal U_1(R[G])\) of units of augmentation one in the group ring \(R[G]\) of \(G\) over the ring of integers \(R\) in \(K\) is hyperbolic in the sense of Gromov.
The investigations naturally lead to consider the unit group \(\mathcal U(H(R))\) of the quaternion algebra \(H(R)\) over \(R\). It is a highly non-trivial task to determine finitely many generators for a subgroup of finite index in this group in case \(H(K)\) is a division ring [see C. Corrales, E. Jespers, G. Leal, Á. del Río, Adv. Math. 186, No. 2, 498-524 (2004; Zbl 1053.16024)]. The authors produce and investigate new units, called Pell and Gauss units in \(H(R)\). As an application of [loc. cit.] and the obtained properties, one obtains a set of generators for the unit group \(\mathcal U(\mathbb{Z}[\sqrt{-7}])\).
The investigations naturally lead to consider the unit group \(\mathcal U(H(R))\) of the quaternion algebra \(H(R)\) over \(R\). It is a highly non-trivial task to determine finitely many generators for a subgroup of finite index in this group in case \(H(K)\) is a division ring [see C. Corrales, E. Jespers, G. Leal, Á. del Río, Adv. Math. 186, No. 2, 498-524 (2004; Zbl 1053.16024)]. The authors produce and investigate new units, called Pell and Gauss units in \(H(R)\). As an application of [loc. cit.] and the obtained properties, one obtains a set of generators for the unit group \(\mathcal U(\mathbb{Z}[\sqrt{-7}])\).
Reviewer: Eric Jespers (Brüssel)
MSC:
| 16U60 | Units, groups of units (associative rings and algebras) |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 11R11 | Quadratic extensions |
| 16S34 | Group rings |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
| 11R27 | Units and factorization |
| 16K20 | Finite-dimensional division rings |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
hyperbolic groups; quaternion algebras; free groups; group rings; units of augmentation one; unit groups; finite sets of generatorsCitations:
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