Reduced Whitehead Groups and the Conjugacy Problem for Special Unitary Groups of Anisotropic Hermitian Forms

Let K / k be a separable field extension of degree 2, D be a finite-dimensional central division algebra over K with a K / k -involution τ , h be an Hermitian anisotropic form on a right D -vector space with respect to τ , and let U ( h ) be the unitary group of h . Then the reduced Whitehead group of its special linear subgroup is defined as follows: , where [ U ( h ), U ( h )] is the commutator subgroup of U ( h ). The first main result establishes a link between the above group and its analog SUK 1 ( h ) for the case of isotropic h (with respect to the same τ ). Theorem . There exists a surjective homomorphism from to SUK 1 ( h ). Furthermore, we also give a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups. Bibliography: 32 titles.