(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