Free Subgroups of Quaternion Algebras
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Using the theory of group actions on trees, we shall prove that if a quaternion algebra over Laurant polynomials is not split then a certain congruence subgroup of the group of norm one elements is a free group. This generalizes and gives an easy, conceptually simpler proof than that given by Pollen for the field of real numbers.
Units, groups of units (associative rings and algebras), Unimodular groups, congruence subgroups (group-theoretic aspects), congruence subgroup, Subgroup theorems; subgroup growth, Finite-dimensional division rings, Laurent polynomials, group actions on trees, free group, quaternion algebra, group of norm one elements, Groups acting on trees
Units, groups of units (associative rings and algebras), Unimodular groups, congruence subgroups (group-theoretic aspects), congruence subgroup, Subgroup theorems; subgroup growth, Finite-dimensional division rings, Laurent polynomials, group actions on trees, free group, quaternion algebra, group of norm one elements, Groups acting on trees
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