We give examples of countable linear groups $\Gamma < \operatorname {SL}_{{\mathbf {3}}}({\mathbf {R}})$, with no nontrivial normal abelian subgroups, that admit a faithful sharply $2$-transitive action on set. Without the linearity assumption, such were recently constructed by Rips, Segev, and Tent in [J. Eur. Math. Soc. 19 (2017), pp. 2895â??2910]. Our are permutational characteristic $2$, se...

We develop a non-compact version of the Hopf maps based on the split-algebras. The split-algebras consist of three species: split-complex numbers, split-quaternions, and split-octonions. They correspond to three non-compact Hopf maps that represent topological mapping between higher dimensional hyperboloids in different dimensions with hyperboloid-bundles. We realize such non-compact Hopf maps ...

‎We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups‎. ‎We prove Arthur's conjecture‎, ‎the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group‎, ‎for quasi-split special unitary groups and their inner forms‎. ‎Furthermore‎, ‎we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms‎. ‎This w...