in this article, by using a partial on locally compact semi-direct product groups, we present a compatible extension of the fourier transform. as a consequence, we extend the fundamental theorems of abelian fourier transform to non-abelian case.

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as dense subgroup in the isometry Urysohn space and automorphism random graph. In fact, we show same for all groups known infinite ultraextensive spaces. These include, addition, rational space, ultrametric spaces, Kn-free graph n≥3. Furthermore, actions on metric spaces or some rel...

Abstract The main result includes as special cases on the one hand, Gerstenhaber–Rothaus theorem (1962) and its generalisation due to Nitsche Thom (2022) and, other Brodskii–Howie–Short (1980–1984) generalising Magnus’s Freiheitssatz (1930).