Survey of the theory of multiple commutators in classical and classical-like groups

29/03/2019, 14:00 - 15:00 in MAP/0G/018

Zhang Zuhong (Beijing Institue of Technology)


Commutators groups play an important role in the structure of classical-like groups and their algebraic K-theory. This talk will survey the theory of mixed commutator groups and multiple mixed commutator groups of congruence and relative elementary groups of classical and classical-like groups. At first only mixed commutators of 2 groups will be considered and it is assumed that at least one of the groups is absolute. We begin the survey with results of C. Jordan on the general linear group over prime fields around 1870, then results for all classical groups over all fields by E. Dickson around 1900, and then results of J. Dieudonne over division rings in the 1940’s, which extend the previous results. Moving into the modern age, we begin with results of W. Klingenberg around 1960 for the general linear, symplectic and even dimensional orthogonal groups over local and semi-local rings and then their generalizations in the 1960’s by H. Bass to the general linear group over rings satisfying a dimension condition (stable rank) and by A. Bak to even dimensional unitary groups over form rings satisfying a dimension condition. These results were then extended by a bunch of people to groups defined over module finite and quasi-finite rings. In 2008, A. Stepanov and N. Vavilov dropped the condition that one of the groups must be absolute and obtained partial results for the general linear group over quasi-finite rings. In 2013, complete results were obtained for arbitrary multiple mixed commutators in the same group over the same rings by H. Hazrat and the speaker. In 2017 these were extended by H. Hazrat, N. Vavilov and the speaker to even dimensional unitary groups over quasi-finite form rings.

Mathematical Sciences Research Centre