These are problems about free groups, their automorphisms and related issues. See also problems (O1), (O7), (O8), (09), (010) in the OUTSTANDING PROBLEMS section.
 

(F1) (a) Is there an algorithm for deciding if a given automorphism of a free group has a non-trivial fixed point ? Background

(b) Is there an algorithm for deciding if a given finitely generated subgroup of a free group is the fixed point group of some automorphism ? Background

*(F2) (H.Bass) Does the automorphism group of a free group satisfy the "Tits alternative" ? Background

*(F3) (V.Shpilrain) If an endomorphism \phi of a free group F of finite rank takes every primitive element to another primitive, is \phi an automorphism? Background

*(F4) Denote by  Orb_{\phi}(u) the orbit of an element  u  of the free group  F_n  under the action of an automorphism \phi. That is,  Orb_{\phi}(u)={v \in F_n,   v=\phi^m(u)  for some  m \ge 0}.  If an orbit like that is finite, how many elements can it possibly have if  u  runs through the whole group  F_n, and  \phi  runs through the whole group  Aut(F_n) ?  Background

(F5) (H.Bass) Is the automorphism group of a free group "rigid", i.e., does it have only finitely many irreducible complex representations in every dimension?   Background

(F6) The conjugacy problem for the automorphism group of a free group of finite rank.  Background

*(F7) (V.Shpilrain) Denote by Epi(n, k) the set of all homomorphisms from a free group F_n onto a free group F_k; n, k \ge 2. Are there 2 elements g_1, g_2 \in F_n with the following property: whenever phi(g_i) = \psi(g_i), i = 1, 2, for some homomorphisms \phi, \psi \in Epi(n, k), it follows that \phi = \psi ? (In other words, every homomorphism from Epi(n,k) is completely determined by its values on just 2 elements). Background

(F8) (W.Dicks, E.Ventura) Let f be an endomorphism of a free group F_n, S a subgroup of F_n having finite rank. Is it true that rank(Fix(f) \cap S) \le rank(S) ? Background

(F9) (A.I.Kostrikin) Let F be a free group of rank 2 generated by x, y. Is the commutator [x,y,y,y,y,y,y] a product of fifth powers in F? (If not, then the Burnside group B(2,5) is infinite).

*(F10) (A.I.Mal'cev) Can one describe the commutator subgroup of a free group by a first order formula in the group theory language ? Background

(F11) (G.Bergman) Let S be a subgroup of a free group F, R a retract of F. Is it true that the intersection of R and S is a retract of S? Background

(F12) (G.Baumslag) Let F = F_n be a free group generated by {x_1, ..., x_n}, and let F^Q be the free Q-group, i.e., the free object of rank n in the category of uniquely divisible groups. Consider a canonical map x_i \arrow (1 + x_i) from F^Q into the formal power series ring \Delta_Q with coefficients in Q. It is known that this map induces a homomorphism \lambda : F^Q \arrow \Delta_Q (Magnus homomorphism). Is \lambda injective? Or, equivalently, is the group F^Q residually torsion-free nilpotent? Background

(F13) (I.Kapovich) Is the group F^Q  in the previous problem linear?  Background

(F14) Let F be a non-cyclic free group of finite rank, and G a finitely generated residually finite group. Is G isomorphic to F if it has the same set of finite homomorphic images as F does?  Background

(F15) (V.Shpilrain) Let F be a non-cyclic free group, and R a non-cyclic subgroup of F. Suppose the commutator subgroup [R, R] is a normal subgroup of F. Is R necessarily a normal subgroup of F ?  Background

*(F16)  (V.Remeslennikov)  Let R be the normal closure of an element  r  in a free group F  with the natural length function,  and suppose that  s  is an element of minimal length in R.  Is it true that  s  is conjugate to one of the following elements:  r,  r^{-1},  [r, f],  or  [r^{-1},  f]  for some element  f ?  Background

*(F17) (V.Shpilrain) Let  u  be an element of a free group F.  We call  u  a strong test element if, whenever  f(u) \ne 1  belongs to the normal closure of  u  for some (injective) endomorphism  f  of the group F,  this   f  is actually an automorphism.  Give a particular example of a strong test element.  Background

(F18) (V.Shpilrain) An automorphism of a group is said to be normal if it leaves invariant every normal subgroup of finite index.  A. Lubotzky and A. S.-T. Lue have proved that every normal automorphism of a free group is inner.  This yields the following question:
Is there a single normal subgroup R of a free group F, so that every automorphism of  F  that leaves R  invariant, is inner?   Background

(F19) (M.Wicks) (a) Let F be a non-cyclic free group of rank n, and P(n,k) the number of its primitive elements of length k. What is the growth of P(n,k) as a function of k, with n fixed ?   Background

(b) The same question for the number of cyclically reduced primitive elements.

(F20) (C.Sims)  Is the c-th term of the lower central series of a free group of finite rank the normal closure of basic commutators of weight c ?  Background

*(F21) (A.Gaglione, D.Spellman) Let F be a non-cyclic free group, and G the Cartesian (unrestricted) product of countably many copies of F. Is the group G/[G,G] torsion-free?  Background

(F22) (P.M.Neumann) Let G be a free product amalgamating proper subgroups H and K of A and B, respectively. Suppose that A, B, H, K are free groups of finite ranks. Can G be simple?  Background

(F23) (A.Olshanskii) Does the free group of rank 2 have an infinite ascending chain of fully invariant subgroups, each being generated (as a fully invariant subgroup) by a single element?

(F24) (A.Miasnikov, V.Remeslennikov)  Let G be a free product of two isomorphic free groups of finite ranks amalgamated over a finitely generated subgroup.
*(a) Is the conjugacy problem solvable in G?
(b) Is there an algorithm to decide if G is free ?
(c) Is there an algorithm to decide if G is hyperbolic?  Background

(F25) (A.Miasnikov, V.Shpilrain)  Let  u  be an element of a free group F_n,  whose length  |u|  cannot be decreased by any automorphism of F_n.   Let  A(u)  denote the set of elements  {v \in F_n;  |v| = |u|,  f(v)=u  for some  f \in Aut(F_n)}.
(a) Is it true that the cardinality of  A(u)  is bounded by a polynomial function of |u| ?
(b) If the free group has rank 2, is it true that the cardinality of  A(u)  is bounded by  c |u|^2  for some constant  c,  which is independent of  u ?  Background

(F26) (M.Bestvina) Let \phi, \psi be two automorphisms of a free group F_n. Is it true that the intersection of Fix(\phi) and Fix(\psi) equals Fix(\alpha) for some automorphism \alpha of F_n ?  Background
 
(F27) (W.Magnus)  Let  u  be an element of a free group F_n.  An element  r  in F_n is called a normal root of  u  if  u  belongs to the normal closure  of  r  in the group  F_n.  Can an element  u, which does not belong to the commutator subgroup [F_n, F_n],  have infinitely many non-conjugate normal roots ?  Background
 
(F28) (S.Sidki) Let S be a subgroup of index 2 in the group F_2, and let R be an isomorphic copy of S (in F_2). Denote by f an isomorphism between S and R. Is there necessarily a non-trivial subgroup H in S which is invariant under f ?   Background
 
(F29) (W. Dicks, E. Ventura) Let H be a subgroup of a free group F_n, and let r(H) denote the rank of H. We call H inert if r(H \cap K) is not bigger than r(K) for any subgroup K of F_n. Is every retract of F_n inert? Background
 
(F30) (W. Dicks, E. Ventura) Let H be a subgroup of a free group F_n, and let r(H) denote the rank of H. We call H compressed if r(H) is not bigger than r(K) for any subgroup K containing H. If H is compressed in F_n, is H necessarily inert? (See the previous problem (F29)). Background

(F31) (J. Stallings) The equalizer of two homomorphisms   \alpha, \beta: F_n \to F_m   is the group   Eq(\alpha, \beta)={x \in F_n : \alpha(x)=\beta(x) }. Is it true that if \alpha is injective, then the rank of   Eq(\alpha, \beta))   is at most n ? Background

(F32) An automorphism of a free group F is called an IA-automorphism if it is Identical on the Abelianization   F/[F, F]. Obviously, all IA-automorphisms form a (normal) subgroup   IA(F)   of the group   Aut(F).  Is the group   IA(F_n)   finitely presented for   n > 3 ? Background

(F33) (A. Casson) Let   f   be an automorphism of F_n. Is it true that there is a subgroup K of finite index in F_n, invariant under   f, such that every eigenvalue of   f_K is equal to a root of 1, where   f_K denotes the induced automorphism of K/[K,K] ?

(F34) (A.Miasnikov, V.Shpilrain)  Let F_n be the free group of a finite rank   n, with generators x_1,...,x_n. An element   u   of F_n is called positive if no x_i occurs in   u   to a negative exponent. An element   u   is called potentially positive if   \alpha(u) is positive for some automorphism \alpha of the group F_n. Finally,   u   is called stably potentially positive if it is potentially positive as an element of F_m for some   m \ge n.
(a) Is the property of being potentially positive algorithmically recognizable?
(b) Are there stably potentially positive elements of F_n that are not potentially positive ? Background

(F35) (J.Wiegold) Let R be a characteristic subgroup of a free group F= F_n. Can F/R be an infinite simple group? Background