FREE GROUPS

These are problems about free groups, their automorphisms and related issues. See also problems (O1), (O7), (O8), (O9), (O10) 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

(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) (a) 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 ?
*(b) (Yu. Merzlyakov) Is the group   IA(F_n)   linear for   n > 2 ? 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] ? Background

(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

(F36) Is the group Out(F_3) linear?   Background

(F37) (V.Bardakov) For any element g of a free group F_n define its primitive length as the minimal k such that g is a product of k primitive elements. Is there an algorithm to determine the primitive length of a given element g of F_n?   Background

(F38) (I.Kapovich, P.Schupp)
(a) Is there an algorithm which, when given two elements   u, v   of a free group F_n decides whether or not the cyclic length of   f(u) equals the cyclic length of   f(v) for every automorphism   f   of the group F_n?
*(b) Call elements with the property alluded to in part (a) translation equivalent, to simplify the language. Is it true that whenever g is translation equivalent to h in F_n and w(x,y) \in F(x,y) is arbitrary, one has w(g,h) translation equivalent to w(h,g) in F_n?
(c) We say that u is boundedly translation equivalent to v if the ratio of the cyclic lengths of   f(u) and   f(v) is bounded away from 0 and from \infty. Is there an algorithm which, when given two elements in a finitely generated free group, decides whether or not they are boundedly translation equivalent?   Background

(F39) (O. Bogopolski)
(a) Is there an algorithm which, when given a finitely generated subgroup S of a free group F and an element g of F, decides whether or not there is an automorphism of F that takes g to an element of the subgroup S?
*(b) The following special case of part (a) is especially attractive: given a finitely generated subgroup S of a free group F, find out whether or not S contains a primitive element of F.   Background

(F40) (V. Shpilrain)
(a) Let u be an element of a free group F_r. Is it true that there is v \in F_r (that depends on u) that cannot be a subword of any cyclically reduced f(u), where f is an automorphism of F_r?
(b) A special case of interest is where u=[x_1, x_2], where x_1 and x_2 are elements of the same free generating set of F_r. Background

(F41) (D. Puder, C. Wu) Suppose u is not a primitive element of F_r. Is it true that the exponential growth rate of the automorphic orbit of u is the square root of 2r-1 ? Background