TY - JOUR

T1 - Nonabelian free subgroups in homomorphic images of valued quaternion division algebras

AU - Rapinchuk, Andrei S.

AU - Rowen, Louis

AU - Segev, Yoav

PY - 2006/11

Y1 - 2006/11

N2 - Given a quaternion division algebra D, a noncentral element e ∈ D × is called pure if its square belongs to the center. A theorem of Rowen and Segev (2004) asserts that for any quaternion division algebra D of positive characteristic > 2 and any pure element e ∈ D × the quotient D × /X(e) of D × by the normal subgroup X(e) generated by e, is abelian-by-nilpotent-by-abelian. In this note we construct a quaternion division algebra D of characteristic zero containing a pure element e ∈ D such that D ×/X(e) contains a nonabelian free group. This demonstrates that the situation in characteristic zero is very different.

AB - Given a quaternion division algebra D, a noncentral element e ∈ D × is called pure if its square belongs to the center. A theorem of Rowen and Segev (2004) asserts that for any quaternion division algebra D of positive characteristic > 2 and any pure element e ∈ D × the quotient D × /X(e) of D × by the normal subgroup X(e) generated by e, is abelian-by-nilpotent-by-abelian. In this note we construct a quaternion division algebra D of characteristic zero containing a pure element e ∈ D such that D ×/X(e) contains a nonabelian free group. This demonstrates that the situation in characteristic zero is very different.

KW - Multiplicative group

KW - Quaternion division algebra

KW - Residue algebra

KW - Valuation

UR - http://www.scopus.com/inward/record.url?scp=33750531431&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-06-08385-7

DO - 10.1090/S0002-9939-06-08385-7

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:33750531431

SN - 0002-9939

VL - 134

SP - 3107

EP - 3114

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 11

ER -