(GA1) (G.Baumslag) Does the  free  Q-group  F^Q  act freely  on some  \Lambda-tree?

(GA2)(G.Baumslag, A.Miasnikov, V.Remeslennikov) Is a finitely generated  group acting freely on a \Lambda-tree,  automatic (biautomatic)?

(GA3) (A.Miasnikov)  Let  G  be a group acting freely on a  \Lambda-tree. Given any finite set of non-trivial elements of  G \ast G,   is there  a homomorphism of   G \ast G   into  G   such that the images of the given elements are also non-trivial?

(GA4) (O.Kharlampovich, A.Miasnikov, V.Remeslennikov) Is the elementary theory of the class of all groups acting freely on a  \Lambda-tree,  decidable?

(GA5) (S.Sidki) Let G be a group such that it contains a subgroup H of index 2 and H admits a homomorphism f:H-->G; we call such a map a 1/2-endomorphims of G. Then this data can be used to construct a (state-closed) representation r of G into the automorphism group of the binary tree where the kernel(r)= < K| K subgroup of H which is both normal in G and f(K) is contained in K>. We call kernel(r) the f-core(H) and say that f is simple provided f-core(H) is the trivial group.
(a) Which groups admit a simple 1/2-endomorphism?
(b) If G admits a simple 1/2-endomorphism, can G be a free abelian group of infinite rank?
(c) If G admits a simple 1/2-endomorphism, can G be a free group of rank k>1? Background