Geometry of the Thompson group (Q6632108)

Let \(E=\mathsf{Th}\) be the sporadic Thompson group found by \textit{J. G. Thompson} [J. Algebra 38, 525--530 (1976; Zbl 0361.20027)] and constructed by \textit{P. E. Smith} [Bull. Lond. Math. Soc. 8, 161--165 (1976; Zbl 0348.20015)]. Maximal subgroups of \(E\) were determined by \textit{S. A. Linton} [J. Lond. Math. Soc., II. Ser. 39, No. 1, 79--88 (1989; Zbl 0676.20009)].\N\NThe author improves the Thompson-Smith construction. As a result, he obtains three remarkable conclusions: \N\NFirst, he obtains a conceptual explanation of the dichotomy between Lie and finite cases in terms of representations of an \((L_{2} (8) :3)\)-subgroup. \N\NSecond, he constructs in \(E\) subgroups \(^{3}\!D_{4} (2) : 3\), \((F_{21} \times L_{3}(2)): 2\), \((G_{2}(3)\times 3):2\) and \(U_{3}(8):6\), he considers the coset geometry of such subgroups and identifies corresponding geometric presentation with that by \textit{G. Havas} et al. [Ohio State Univ. Math. Res. Inst. Publ. 8, 193--200 (2001; Zbl 0994.20014)]. Thanks to these results he provides a new geometric construction and uniqueness proof for \(E\). \N\NThirdly, he provides a self-contained construction of the Dempwolff-Thompson orthogonal decomposition on the \(E_{8}\)-Lie algebra, along with the stabiliser \(2^{5+10} \cdot L_{5} (2)\) (the so-called Dempwolff group: see [\textit{U. Dempwolff}, Rend. Sem. Mat. Univ. Padova 48(1972), 359--364 (1973; Zbl 0275.20017)]) of this decomposition in the Lie group \(E_{8}(\mathbb{C})\).