On amenable subgroups of $F$-groups

When studying the Banach–Tarski paradox, John von Neumann (1929) introduced the concept of amenable group: a group $G$ is amenable, if it has a left invariant nontrivial finitely additive measure, i.e. a non-negative valued function $\mu$ defined on the set $P(G)$ of all subsets of the set $G$ satisfying

• $\mu (G) \, > \, 0$,
• for all non-intersecting subsets $U$ $V$ of the set $G$ the equality $ \mu (U \cup V) \, = \, \mu (U ) \, + \,\mu ( V) $ holds,
• for any subset $U$ of the set $G$ and for all element $g$ of the group $G$ the equality $ \mu (g U ) = \mu (U ) $ holds.

John von Neumann (1929) found that any locally solvable group is amenable, and any free non-cyclic group is non-amenable. Since a subgroup of an amenable group is amenable itsef, then any group with an embedded free group of rank 2, is non-amenable. A hypothesis going back to this John von Neumann (1929) work, consists in amenablility of any group in which no free group of rank 2 can be emedded.
This leads to the concept of von Neumann alternative for a class $C$ of groups:
for a class $C$ of groups von Neumann alternative for amenability is valid, if for an arbitrary group $G$ from this class the following statement holds:
A group $G$ is either amenable or it contains a subgroup isomorphic to a free $F_2$ group of rank 2.
The original J. von Neumann hypothesis can be considered as von Neumann alternative for amenability for the class of all groups. The von Neumann alternative for amenability holds for the class of subgroups of groups with one definig relation as well as for the class of all groups satisfying small cancellation conditions.
In this work we establish the validity of von Neumann alternative for amenability of subgroups of $F$-groups. The following equivalence is shown for an arbitrary subgroup $G$ of any $F$-group:
A group $G$ is either amenable or it contains a subgroup isomorphic to a free $F_2$ group of rank 2.
Bibliography: 15 titles.