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

(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) (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