We prove the Freiheitssatz for free associative algebras over any field. This extends the previous result of Marker-Limanov and immediately implies the Freiheitssatz for free restricted Lie algebras and free Lie algebras. We prove the Freiheitssatz for free Poisson algebras over a field of characteristic zero. This answers a question of MarkerLimanov and Umirbaev in the affirmative.

It is proven that the Freiheitssatz holds for all one-relator products of cyclic groups if the relator is cyclically reduced and a proper power. The method of proof involves representing such groups in PSL2(C) and is a refinement of a technique of Baumslag, Morgan and Shalen. The technique allows the extension of the Freiheitssatz result to many additional one-relator products.

In this paper, we generalise Magnus’ Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.

the freiheitssatz of magnus for one-relator groups is one of the cornerstones of combinatorial group theory. in this short note which is mostly expository we discuss the relationship between the freiheitssatz and corre-sponding results in free power series rings over fields. these are related to results of schneerson not readily available in english. this relationship uses a faithful representa...

We prove the Freiheitssatz for right-symmetric algebras and the decidability of the word problem for right-symmetric algebras with a single defining relation. We also prove that two generated subalgebras of free right-symmetric algebras are free and automorphisms of two generated free right-symmetric algebras are tame. Mathematics Subject Classification (2000): Primary 17A36, 17A50; Secondary 1...

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).