(OR1) For a background to this problem, see the survey [G.Baumslag, Some open problems. Summer School in Group Theory in Banff, 1996, 1--9. CRM Proceedings and Lecture Notes. 17. Amer. Math. Soc., Providence, 1999].
(OR2) For some partial results, see the background to the problem (O4).
(OR5) J.Harlander [Solvable groups with cyclic relation module, J. Pure Appl. Algebra 90 (1993), 189--198] showed that the answer is "yes" in the case where $G$ is finitely generated and solvable.
(OR7) (b), (c) A solution of these two problems was communicated to us by A.Olshanskii. In fact, the commutator subgroup [F,F] here can be replaced by ANY non-cyclic subgroup of a free group F; the answer will still be negative. It follows from a result of [A.Olshanskii, SQ-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119--132] that for any m, every non-cyclic subgroup H of F contains a subgroup K, which is a free group of rank m, with the following property: for any normal subgroup U of K, the normal closure of U in F intersects K by the group U. To apply this result to our situation, take two elements, x and y, that generate a subgroup K=F_2 of H = [F,F] with the property described above. Let r be a Baumslag-Solitar relator built on these two elements; for example, take r = xyx^{-1}y^{-2}. Let U be the normal closure (in K) of r. Then, from what is said in the previous paragraph, it follows that the normal closure of U in F (call it V) intersects K by U. Therefore, the (one-relator) group F/V contains a subgroup KV/V which is isomorphic to a Baumslag-Solitar group, hence F/V can be neither residually finite nor automatic.
(OR8) This problem, as well as (OR7)(a), is motivated by the desire to find a non-hopfian one-relator group which is essentially different from any of the Baumslag-Solitar groups [G.Baumslag, D.Solitar, Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc. 68 (1962), 199--201].
(OR10) Note that hyperbolic
groups are automatic, and, in particular, an amalgamated product of two
free groups with finitely generated subgroups amalgamated is hyperbolic
if at least one of the subgroups is malnormal [O.Kharlampovich and A.Myasnikov,
Hyperbolic groups and free constructions. Trans. Amer. Math. Soc.
350 (1998), 571--613].
Furthermore, an amalgamated product of two finitely generated
abelian groups is automatic [G.Baumslag, S.M.Gersten, M.Shapiro, H.Short,
Automatic groups and amalgams -- a survey. Algortims and Classification
in Combinatorial Group Theory (Berkeley, CA, 1989), 179--194, Math.
Sci. Res. Inst. Publ. 23. Springer, New York, 1992].
(OR14)
We note that recently, O.Kharlampovich and A.Miasnikov [Irreducible affine
varieties over groups, J.Algebra 200 (1998), 517-570] proved that every
finitely generated group which is discriminated by a free group can be
obtained from a free group by applying finitely many free constructions
of a very particular type.
(FP2) We note that there are infinitely presented (but finitely generated) groups with this property - see [J.M.T.Jones, Direct products and the Hopf property. J. Austral. Math. Soc. 17 (1974), 174--196]. Moreover, the same author has constructed, for any $n \ge 2$, a (infinitely presented) group $G$ isomorphic to its $n$th direct power $G^n$, but non-isomorphic to $G^k$ for any $k, 1<k<n$ -- see [J.M.T.Jones, On isomorphisms of direct powers. Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), 215--245, Stud. Logic Foundations Math., 95, North-Holland, 1980].
(FP3) A paper by M.Bestvina, N.Brady with a negative solution of this problem has been published recently [Invent. Math. 129 (1997), 445--470]. Explicit presentations of their groups have been given by W.Dicks and I.Leary [Presentations for subgroups of Artin groups, Proc. Amer. Math. Soc. 127 (1999), 343--348].
(FP4) It is easy to see that the answer is affirmative if the group has rank 2; this is due to the fact that in the free group of rank 2, any element is in the normal closure of a primitive element.
(FP7) It is -- see [O.Kharlampovich, A.Myasnikov, Irreducible affine varieties over groups. I, II. J.Algebra 200 (1998), 472--516, 517--570].
(FP8) It is; see the preprint [M.Feighn, M.Handel, Mapping tori of free group automorphisms are coherent] at http://andromeda.rutgers.edu/~feighn/
(FP14) No, it is not; see a recent preprint [J. Howie, A proof of the
Scott-Wiegold conjecture on free products of cyclic groups] at
http://www.ma.hw.ac.uk/~jim/
(FP15)
A single element g \in G is called a test element
(see [V. Shpilrain, Recognizing automorphisms of the
free groups, Arch. Math. 62 (1994), 385--392])
if, whenever \phi(g)=g for some endomorphism \phi of the group G,
this \phi is an automorphism of G. Thus, if G has
a test element, the test rank of G is 1. For example, any free group
of finite rank has test rank 1. On the other hand, there are groups
(for example, free abelian groups of finite rank) whose test rank equals
their rank. (Obviously, it cannot be bigger than that.)
(FP18) R.Hirshon
himself [Some properties of endomorphisms in residually finite groups,
J. Austral. Math. Soc. Ser. A 24 (1977), 117--120 ] proved that in
the case where f(G) has finite index in G. However, the answer is negative
in general [D.Wise, A continually descending endomorphism of a finitely
generated residually finite group, Bull. London Math. Soc. 31 (1999),
45--49].
(FP19) J.Makowsky [On some conjectures connected with complete
sentences,
Fund. Math. 81 (1974), 193-202] pointed out that the
affirmative
answer to this problem would give an example of a complete finitely
axiomatizable theory T which is categorical in uncountable cardinals
but not $\omega$--categorical. Subsequently, examples of such
theories were given by Peretyat'kin (1980) and others
(see [Hodges, Model theory, p. 619]).
(B1), (B2) There
are two canonical representations of braid groups by matrices over Laurent
polynomial rings - the Burau and Gassner representations (the latter is
actually a representation of the pure braid group which is a subgroup of
finite index in the whole braid group). Both of these representations are
faithful for n = 2,3 (a general reference here is [J.S.Birman, Braids,
links and mapping class groups, Ann. Math. Studies 82, Princeton Univ.
Press, 1974]). A proof of the Gassner representation being faithful for
every n (which implies braid groups being linear) was recently claimed
in [S.Bachmuth, Braid groups are linear groups, Adv. Math. 121 (1996),
50--61]. However, there is a controversy around this paper since several
people believe they have found essential gaps in the proof (see J. S. Birman's
review article 98h:20061 in Math. Reviews). This makes us consider
Problem (B2) open.
(B3) The
Burau representation was shown to be non-faithful for n \ge 10 in [J.Moody,
The faithfulness question for the Burau representation, Proc. Amer. Math.
Soc. 119 (1993), 671--679], and then for n \ge 6 in [D.Long and M.Paton,
The Burau representation is not faithful for n \ge 6, Topology 32 (1993),
439-447]. More recently, S.Bigelow [The Burau
representation is not faithful for n=5, Geom. Topol. 3 (1999),
397--404 (electronic)]
has shown that
the answer is
negative for n = 5 as well.
(B5), (B6), (B7) For
a background and discussion on these problems, we refer to a recent preprint
by V.Lin on "Braids, permutations, polynomials.I" which can be either found
on the Max Planck Institut f\"{u}r Mathematik electronic preprint server,
or requested from the author at vlin@techunix.technion.ac.il
(B12) The groups B_3 and B_4 do have non-elementary
hyperbolic factor groups (note that B_4 maps onto B_3, but, say,
B_5 does not map onto B_4).
(G1) The
point here is that there are examples of groups of intermediate growth
(between polynomial and exponential), but all these groups are infinitely
presented -- see [R.I.Grigorchuk, On the Milnor problem of group growth.
(Russian) Dokl. Akad. Nauk SSSR 271 (1983), 30--33]; [R.I.Grigorchuk,
Construction
of p-groups of intermediate growth that have a continuum of factor-groups.
(Russian) Algebra i Logika 23 (1984), 383--394, 478]; [R.I.Grigorchuk,
Degrees of growth of p-groups and torsion-free groups. (Russian) Mat.Sb.
126(168) (1985), 194--214, 286]; [R.I.Grigorchuk, A.Maki, On a group of
intermediate growth that acts on a line by homeomorphisms. (Russian) Mat.
Zametki 53 (1993), 46--63; translation in Math. Notes 53 (1993),
146--157].
(G3) A group G has uniformly exponential growth if
there is e>0 such that the growth rate of G with
respect to any generating set is > 1+e.
(G8) R. Sharp [Local limit theorems for free groups,
Math. Ann. 321 (2001), 889--904] showed that the asymptotics of f(n)
is C (2r-1)^n / n^{r/2} for some constant C.
(E2) It
is clear from considering abelianization that, if $G=F_n/R$ is a counterexample,
then $G$ must be perfect, i.e., $G=[G,G]$.
If, in the above notation, the sum of exponents on $x_{n+1}$
in $s$ is not 0, does the equation $s=1$ always have a solution over
$G$?
For recent results on the latter problem, we refer to [A.A.Klyachko,
A funny property of sphere and equations over groups. Comm. Algebra 21
(1993), 2555--2575] and [A.Clifford, R.Z.Goldstein,
Tesselations of $S\sp 2$ and equations over torsion-free groups. Proc.
Edinburgh Math. Soc. (2) 38 (1995), 485--493].
(E3) The answer
was recently shown to be negative [T.Coulbois , A.Khelif, Equations
in free groups are not finitely approximable, Proc. Amer. Math. Soc. 127
(1999), 963-965].
(E5) R.Bryant [The verbal
topology of a group. J.Algebra {\bf 48} (1977), 340--346] and V.Guba [Equivalence
of infinite systems of equations in free groups and semigroups to finite
subsystems. Math. Notes USSR 40 (1986), 688--690] proved
that free groups are equationally noetherian. See also [J.Stallings,
Finiteness properties of matrix representations. Ann. of Math. (2) 124
(1986), 337--346].
(A4) We just note here that
importance of this problem is enhanced by a well-known fact that the knot group of a knot K
is (infinite) cyclic if and only if K is isotopic to the unknot.
(H1) I.Kapovich
and D.Wise [The equivalence of some residual properties of word-hyperbolic
groups, J. Algebra 223 (2000), 562--583] proved the equivalence of (a) and (b).
(H3) We note that Sela
[Ann. of Math. (2) 141 (1995), 217--283] has solved the isomorphism problem
for torsion-free hyperbolic groups that do not split (as an amalgamated
product or an HNN extension) over the trivial or the infinite cyclic group.
(H5)
Papasoglu [An algorithm detecting hyperbolicity. Geometric and computational
perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ,
1994), 193--200, DIMACS Ser. Discrete Math. Theoret. Comput. Sci.,
25,
(H6)-(H10)
For a background on problems (H6) through (H10) we refer to [S. M.Gersten,
Problems on automatic groups. Algorithms and classification in combinatorial
group theory (Berkeley, CA, 1989), 225--232, Math. Sci. Res. Inst. Publ.,
23, Springer, New York, 1992.]
(H12)
For a background on equationally noetherian groups, we refer to
[G.Baumslag, A.Myasnikov, V.Roman'kov, Two theorems about
equationally Noetherian groups, J. Algebra 194 (1997), 654--664.]
(H14)
We note that malnormality is decidable in free groups --
see [G.Baumslag, A.Myasnikov, V.Remeslennikov,
Malnormality is decidable in free groups, Internat. J. Algebra Comput. 9 (1999), 687--692].
(N1) V.Bludov has communicated
the following example of a non-trivial element g of a
free nilpotent group of rank 2 and nilpotency class
4k \ge 8, which is fixed by every automorphism:
(N2) Let $P = \langle
X \mid R \rangle$ be a finite presentation of a group $G$, $F(X)$ a free
group on $X$, and $ncl(R)$ the normal closure of $R$ in $F(X)$. The ``area"
of $w \in ncl(R)$ is defined by $$A(w) = min\{\ m \ \mid \ w = \prod_{i
= 1}^{m}c_i^{-1}r_i^{e_i}c_i , \ c_i \in F(X), r_i \in R, e_i = \pm 1\}.$$
Now, the {\em isoperimetric function} of the presentation $P$ is given
by $$ \Phi_P(n) = max\{\ A(w) \ \mid \ w \in ncl(R), \ |w| \leq n \}, $$
where $|w|$ is the length of $w$ in $F(X).$ Let $N$ be the set of all non-negative
integers. For functions $f,h : N \rightarrow N$ we define a relation $f
\preceq h$ iff there exists a constant $K$ such that $f(n) \leq K \cdot
h(Kn) + Kn$ for every $n \in N.$ We write $f \simeq h$ iff $f \preceq h$
and $h \preceq f$. It is not hard to show that if $P$ and $Q$ are two finite
presentations of a group $G$, then $\Phi_P \simeq \Phi_Q$. Any function
equivalent to $\Phi_P$ is called the {\em Dehn function } of $G$. From
now on, we shall denote the Dehn function of a group $G$ by $\Phi_G$. S.Gersten
[{\it Isodiametric and isoperimetric inequalities in group extensions}.
Preprint, University of Utah, 1991] proved that for any finitely generated
nilpotent group $G$, $\Phi_G$ is bounded by a polynomial of degree $2^h$,
where $h$ is the Hirsch length of $G$. G.Conner [{\it Central extensions
of word hyperbolic groups satisfy a quadratic isoperimetric inequality},
Arch. Math. {\bf 65} (1995), 465--479] improved the bound on the degree
to $2^c$, where $c$ is the nilpotency class of $G$. Recently, C.Hidber
[{\it Isoperimetric functions of finitely generated nilpotent groups and
their amalgams}, Ph.D. thesis] proved that $\Phi_G \preceq n^{2c}$. It
is known that if $G$ is a free nilpotent group of class $c$, then $\Phi_G
\simeq n^{c+1}$, in particular, $\Phi_G$ is equivalent to a polynomial.
Whether or not this is true in general, is still an open problem.
(N4)
Hamkins [Every group has a terminating transfinite automorphism
tower, Proc. Amer. Math. Soc. 126 (1998), 3223--3226] established
the property in the title.
(M1)
There is an algorithm to
determine whether or not a given finitely generated metabelian group is
free metabelian -- see [J.R.J.Groves, C.F.Miller, III,
Recognizing free metabelian groups,
Illinois J. Math. 30 (1986), 246--254] and the paper by Noskov
cited in the background to the problem (F14).
(M2) The automorphism
group of a free metabelian group of finite rank is known to be finitely
generated unless the rank equals 3 -- see [S.Bachmuth, H.Mochizuki,
${\rm Aut}(F)\to{\rm Aut}(F/F")$ is surjective for free group $F$ of rank
$\geq 4$, Trans. Amer. Math. Soc. 292 (1985), 81--101]
and [S.Bachmuth, H.Mochizuki, The nonfinite generation of ${\rm
Aut}(G)$, $G$ free metabelian of rank $3$, Trans. Amer. Math. Soc. 270
(1982), 693--700.]
(M4) For groups of finite
rank, the answer is affirmative -- see [V. A.Artamonov, Projective
metabelian groups and Lie algebras. (Russian) Izv. Akad. Nauk SSSR Ser.
Mat. 42 (1978), 226--236, 469. ]
(M5)
For a survey on homological properties of metabelian groups, we refer to
[Yu.V.Kuz'min, Homology theory of free abelianized extensions,
Comm. Algebra 16 (1988), 2447--2533].
(M6) For a background,
we refer to [V.Shpilrain, Fixed points of endomorphisms of a free
metabelian group, Math. Proc. Cambridge Philos. Soc. 123 (1998),
77--85.]
(M7) No, there is no
such group - see [R.Goebel, A.Paras, Outer automorphism groups of metabelian
groups, J. Pure Appl. Algebra 149 (2000), 251--266.]
(S1) The automorphism
group of a free solvable group of derived length > 2 and rank > 2 cannot
be generated by elementary Nielsen automorphisms -- see [C. K.Gupta,
F. Levin, Tame range of automorphism groups of free polynilpotent groups,
Comm. Algebra 19 (1991), 2497--2500] and
[V.Shpilrain, Automorphisms of $F/R'$ groups,
Internat. J. Algebra Comput. 1 (1991), 177--184. ] Moreover, every
free solvable group of derived length d > 2 and rank r> 2
has automorphisms that cannot be lifted to automorphisms of the free solvable
group of derived length d+1 and the same rank r -- see
[V.Shpilrain,
Non-commutative determinants and automorphisms of groups, Comm. Algebra
25 (1997), 559--574.]
(S2) We note that the
word problem for groups admitting finitely many defining relations
in the variety of all solvable groups of a given derived length >2,
is, in general, unsolvable - see [O. G. Kharlampovich, A finitely
presented solvable group with unsolvable word problem. (Russian) Izv. Akad.
Nauk SSSR Ser. Mat. 45 (1981), 852--873, 928.]
(S3) E.Timoshenko [Center
of a group with one defining relation in the variety of $2$-solvable groups
(Russian), Sibirsk. Mat. Z. 14 (1973), 1351--1355, 1368] settled
this problem in the affirmative for metabelian groups. C. K.Gupta
and V.Shpilrain [The centre of a one-relator
solvable group, Internat. J. Algebra Comput. 3 (1993), 51--55]
settled the problem (also in the affirmative) for solvable groups of arbitrary
derived length, under an additional assumption that the relator is not
a proper power modulo any term of the derived series.
(S4) The answer is "yes"
for free metabelian groups -- see [Kh. S.Allambergenov, V.A.Romankov,
Products of commutators in groups (Russian), Dokl. Akad. Nauk UzSSR 1984,
14--15] and for free solvable groups of derived length 3 -- see [A. H.
Rhemtulla, Commutators of certain finitely generated soluble groups.
Canad. J. Math. 21 (1969), 1160--1164.]
(S9) The most obvious
candidate for a test element in a group generated by x and y, would
be u=[x,y]. This however is not a test element in a
free solvable group of derived length d>2 - see [N.Gupta, V.Shpilrain,
Nielsen's commutator test for two-generator groups, Math. Proc. Cambridge
Philos. Soc. 114 (1993), 295--301. ]
(GA5) For a general setup
that motivated this problem, we refer to a recent preprint [V. Nekrashevych, S. Sidki,
Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms],
which is available here.
E. I. Timoshenko [Test elements and test
rank of a free metabelian group, Siberian Math. J. 41 (2000), 1200--1204]
proved that a free
metabelian group of rank > 2 has test rank 2.
C.F.Rocca Jr. and E.Turner have shown that for any pair of integers
(k, n) with 1 \le k \le n, there are finitely generated abelian groups
of rank n and test rank k. Their preprint is available
here.
We also note that S.V.Ivanov has constructed examples,
for big numbers p, of finitely generated (but infinitely presented)
infinite groups of period p with precisely
p conjugacy classes. These examples are included as Theorem 41.2 in
[A.Yu.Ol'shanskii, Geometry of defining relations in groups.
Mathematics and its Applications (Soviet Series), 70.
Kluwer Academic Publishers Group, Dordrecht, 1991].
Problem (B1) has been recently settled in the affirmative by
S.Bigelow [Braid groups are linear, J. Amer. Math. Soc. 14 (2001),
471--486] and D. Krammer [Braid groups are linear, Ann. of Math., to
appear],
who proved that the Krammer representation of the braid group B_n is faithful for
every
n, and therefore all braid groups are linear. For n=4, see also
[D. Krammer, The braid group B_4 is linear,
Invent. Math. 142 (2000), 451--486].
On the other hand, it is known that the Burau representation
is faithful for n = 3 [W.Magnus, A.Peluso, On a theorem of V. I.
Arnold. Comm. Pure Appl. Math. 22 (1969), 683-692].
Here we only note that automorphisms of braid groups were described
in [J.Dyer, E.Grossman, The automorphism groups of the braid groups,
Amer. J. Math. 103 (1981), no. 6, 1151--1169. ]
Recently, S.Humphries [Torsion-free quotients of braid groups.
Internat. J.
Algebra Comput. 11 (2001), 363--373] has constructed a
representation
of the group B_n which is shown to provide torsion-free non-abelian factor
groups of B_n as well as of the commutator subgroup [B_n, B_n]
for n < 7. It is likely that the same representation should work
for other values of n as well.
(B11) An affirmative
answer to this problem would also imply a solution of the problem
(B6).
We also note that the pure braid group P_n can be mapped
onto
the free group F_2 for any n > 2.
This problem was settled affirmatively for the following
classes of groups:
- hyperbolic groups
[M. Koubi, Croissance uniforme dans les groupes
hyperboliques, Ann. Inst. Fourier 48 (1998),
1441--1453]
- one-relator groups
[R. I. Grigorchuk, P. de la Harpe, One-relator groups of
exponential growth have uniformly exponential growth, Math.
Notes 69 (2001), 628--630]
- solvable groups [D.V. Osin, The entropy of solvable groups,
Ergodic Theory and Dynam. Sys., to appear]
Also, it suffices to consider the case where $G$ is an infinite simple
group.
There are several related problems about (systems of) equations
over groups. We only give one of them here; it appears as Problem 2a on
Lyndon's list [R.Lyndon, Problems in combinatorial group theory.
Combinatorial group theory and topology (Alta, Utah, 1984), 3--33, Ann.
of Math. Stud., 111, Princeton Univ. Press, 1987]:
For other related problems, we refer to [J.Howie, How
to generalize one-relator group theory. Combinatorial group theory and
topology (Alta, Utah, 1984), 53--78, Ann. of Math. Stud., 111,
Princeton Univ. Press, 1987].
For a general discussion on this and related problems, we refer
to [G.Baumslag, A.Myasnikov, V.Remeslennikov, Algebraic geometry over groups
I: Algebraic sets and ideal theory, J. Algebra 219 (1999), 16--79].
See also [ W.Whitten, Knot complements and groups,
Topology 26 (1987), 41--44] for a generalization of this fact implying,
in particular, that the knot group of a
prime knot determines the knot type up to the mirror reflection.
(MA2) M.Evans [Primitive
elements in the free metabelian group of rank 3, J.Algebra 220 (1999), 475--491]
has recently found such matrices. It was previously known due to
[S.Bachmuth, H.Mochizuki, $E\sb{2}\not={\rm SL}\sb{2}$ for
most Laurent polynomial rings. Amer. J. Math. 104 (1982), 1181--1189]
that such matrices do exist.
Amer. Math. Soc., Providence, RI, 1996] gave
a partial algorithm to recognize hyperbolic groups. Given a finite presentation
$\langle S,R\rangle$, the algorithm terminates if the group $G=\langle
S,R\rangle$ is hyperbolic and gives an estimate of the hyperbolicity constant
$\delta$.
Recently, M.Bridson and D.Wise
[Malnormality is undecidable in hyperbolic groups] have settled the general case in the
negative. Their preprint is available
here.
g = [a,[a,b],[a,b,b], [a,b],...,[a,b]], where there are
(2k-3) occurrences of [a,b] after [a,b,b].
(Here a and b are generators of the free nilpotent
group).
Recently, A. Papistas [A note on fixed points of certain relatively
free nilpotent groups, Comm. Algebra 29 (2001), 4693--4699]
and, independently,
E.Formanek [Fixed Points and Centers of Automorphism Groups
of Free Nilpotent Groups, Comm. Algebra, to appear] have solved this problem
completely by
classifying all pairs (r,c) for which
F(r,c), the free nilpotent group of rank r and class c,
has nontrivial elements fixed by all automorphisms. Formanek's preprint is
available
here.
We also note
that "most" algorithmic problems about finitely presented metabelian groups
have solutions by now - see [G.Baumslag, F.Cannonito, D.Robinson,
The algorithmic theory of finitely generated metabelian groups, Trans.
Amer. Math. Soc. 344 (1994), 629--648] and references thereto.
The problem was recently answered in the negative by M. Kassabov;
his preprint is available
here.
It is not known however whether
or not those automorphism groups are finitely
generated.
Recently, V.Roman'kov has constructed test elements in the
free solvable group of rank 2 and derived length 3 - see his
preprint.
It is plausible that the same method can be used for constructing test
elements in the free solvable group of any bigger rank as well,
but technically it is getting more complicated.
We also mention a related result of E.I.Timoshenko [Test
elements and test rank of a free metabelian group, Sib. Mat. Zh. , to appear]
who proved that a free metabelian group of rank > 2 does NOT have any test
elements. It was previously known [V.G.Durnev, The Mal'tsev-Nielsen equation
on a free metabelian group of rank 2. Math. Notes USSR 46 (1989), 927--929]
that the free metabelian group of rank 2 does have test elements,
for example, u=[x,y].