Groups generated by k-root subgroups – a survey

Abstract Root Subgroups and Quadratic Action

“…(We neglect the maximality condition of [Ti1], [Ti2] which was originally a part of the definition. If (A, B) & (P) SL 2 (k) it follows from [Ti1,(2.1)], that II( ;) is satisfied.…”

“…But to have a uniform notation, i.e., to be able to speak of (Siegel)-transvections we call them Siegel-transvections. For definition of Siegel-transvections see ([Ti2,(9.1), (9.2)]). P0(V, q) is the group induced by 0(V, q) on the projective space^(V ).…”

Developments in finite Phan theory

“…One approach is also based on the construction of BN-pairs, see [132], while an alternative approach (see [128,131,133]) is based on his theory of abstract root subgroups [127,129,130]; see Section 4.4. [105], Humphreys [88], Timmesfeld [132], Dunlap [54]).…”

“…A completely different and independent approach to the Curtis-Tits Theorem is based on the classification of groups generated by a class of abstract root subgroups [127,129,130]. This wonderful classification result makes it possible to prove all sorts of generalisations of Steinberg-presentation-type results and the Curtis-Tits Theorem, see [128,131,133].…”

“…Among the most noteworthy and influential of these advances are the systematic application of the concept of amalgams based on [49,59,117], the local-to-global approach [136], ingenious applications of combinatorial topology and geometric group theory (as in [2,91,92,102,137]), the theory of abstract root groups [127,129,130], and the interaction of Kac-Moody groups and twin buildings [34,35,36,39,38,138,140]. These methods have proven fruitful over and over again in proving, simplifying and generalising several results in group theory and have had their impact in other areas of mathematics.…”