NILPOTENT GROUPS

*(N1) (A.Miasnikov) Let G be a free nilpotent group of finite rank. Suppose an element g \in G is fixed by every automorphism of G. Is it true that g = 1 ?  Background

*(N2) Let G be a finitely generated nilpotent group. Is the Dehn function of G equivalent to a polynomial? Background

(N3) (B.I.Plotkin) Is it true that every locally nilpotent group is a homomorphic image of a torsion-free locally nilpotent group?

(N4) (G.Baumslag) Let G be a finitely generated torsion-free nilpotent group. Is it true that there are only finitely many non-isomorphic groups in the sequence Aut(G), Aut(Aut(G)), ... ? Background

(N5) (G.Baumslag) Is the property of being directly indecomposable decidable for finitely generated nilpotent groups?

(N6) (A.Miasnikov) Describe all finitely generated nilpotent groups of class 2 which have genus 1. (We say that a group G has genus 1, if every group with the same set of finite homomorphic images as G, is isomorphic to G).

(N7) Is every group with an Engel identity [x,y,...,y] = 1, locally nilpotent?   Background

(N8) (A.Miasnikov) * (a) Is it true that equations of the form [x,y] = g (g \in G) are decidable in every finitely generated 2-nilpotent group G?
(b) Are equations of the form [x,y] = g (g \in G) decidable in every finitely generated free nilpotent group?   Background

(N9) (A.Miasnikov) Let G be a group. The retract problem in G is the following: given a finitely generated subgroup H of G, decide if H is a retract of G or not.
(a) Is the retract problem decidable in every finitely generated nilpotent group G?
*(b) Is the retract problem decidable in finitely generated free nilpotent groups?   Background