13.39 (1995)

Let $A$ be an associative ring with unity and with torsion-free additive group, and let $F^A$ be the tensor product of a free group $F$ by $A$ (A. G. Myasnikov, V. N. Remeslennikov, Siberian Math. J., 35, no. 5 (1994), 986–996); then $F^A$ is a free exponential group over $A$; in (A. G. Myasnikov, V. N. Remeslennikov, Int. J. Algebra Comput., 6 (1996), 687–711), it is shown how to construct $F^A$ in terms of free products with amalgamation.
$\qquad$ a) (G. Baumslag). Is $F^A$ residually nilpotent torsion-free?
$\qquad$ b) Is $F^A$ a linear group?
$\qquad$ c) (G. Baumslag). Is the Magnus homework of $F^\mathbb{Q}$ into the group of power series over the rational number field $\mathbb{Q}$ faithful or not?
$\qquad$ d) Is the universal theory of $F^A$ decidable?
$\qquad$ e) (G. Baumslag). Can free $A$-groups be characterized by a length function?
$\qquad$ f) (G. Baumslag). Does a free $\mathbb{Q}$-group admit a free action on some $\Lambda$-tree? See definition in (R. Alperin, H. Bass, in: Combinatorial group theory and topology, Alta, Utah, 1984 (Ann. Math. Stud., 111), Princeton Univ. Press, 1987, 265–378).