Nonabelian free subgroups in homomorphic images of valued quaternion division algebras

Andrei S. Rapinchuk, Louis Rowen, Yoav Segev

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)3107-3114
Number of pages8
JournalProceedings of the American Mathematical Society
Volume134
Issue number11
DOIs
StatePublished - 1 Nov 2006

Keywords

  • Multiplicative group
  • Quaternion division algebra
  • Residue algebra
  • Valuation

Fingerprint

Dive into the research topics of 'Nonabelian free subgroups in homomorphic images of valued quaternion division algebras'. Together they form a unique fingerprint.

Cite this