problems_incl.tex from Magnus at Krugle
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\appendix
\chapter{Sample List of Open Problems}
\label{problems}
\begin{enumerate}
\item Let G be a free product amalgamating proper subgroups H and K of
A and B, respectively.
\noindent a) Suppose that H, K are finite and
$\mid A:H\mid > 2, \mid B:K\mid \geq 2$.
Is G SQ-universal?
\noindent
b) Suppose that A, B, H, K are free groups of
finite ranks. Can G be simple?
\medskip\noindent
P. M. Neumann
\item Show that the growth function $f$ of any infinite finitely
generated group satisfies the inequality
$$f(n) \leq (f(n-1)+f(n+1))/2 $$
for all sufficiently large $n$ (with a fixed finite system of generators).
\medskip\noindent
V.V. Beliayev, N.S. Sesekin
\item It is not hard to show that a finite perfect group is
the normal closure of a single element. Is the
same true for infinite finitely generated groups?
\medskip\noindent
J. Wiegold
\item
Prove or disprove that for all, but finitely many, primes p the group
$$G_p+<a,b;a^2=b^p=(ab)^3=(b^rab^{-2r}a)^2=1>,$$
where $r^2+1\cong 0 (mod p)$, is infinite. A solution of this problem
would have interesting topological
applications. It was proved with the aid of computer that G
is finite for $p \leq 17$.
\medskip\noindent
J. Mennicke
\item
For what n and what group words w, the defining relations of the group
$$<a,x;a^n=1, w(a,x)=1>$$
imply that $a=1$? In particular, is it true that $a \neq 1$ in
$<a,x; a^5=1, a^{x^2}=[a,a^x]>$?
\medskip\noindent
Lyndon
\item
Let
$$G(a,b)=<x,y;x=[x,_ay], y=[y,_bx]>.$$
Is $G(a,b)$ finite? It is easy to show that
$G(1,b)=1$ and one can show that
$G(2,2)=1$. Nothing is known about $G(2,3)$.
If one could show that every minimal simple
group is a quotient of some $G(a,b)$, then this would yield a
very nice sequence of words in two variables
to characterize solvable groups (see R. Brandl, J.S. Wilson,
J. Algebra, 116 (1988), 334-341.)
\medskip\noindent
Brandl
\item
Suppose that we have $[a,b]=[c,d]$ in an absolutely free group,
where $a,b,[a,b]$ are basic commutators (in some fixed set of
free generators). If $c$ and $d$ are arbitrary (proper) commutators, does
it follow that $a=c$ and $b=d$?
\medskip\noindent
A. Gaglione, D. Spellman
\item
Is the commutator $[x,y,y,y,y,y,y]$ a product of fifth powers in a
free group $< x,y>$? If not, then the
Burnside group $B(2,5)$ is infinite.
\medskip\noindent
A.I. Kostrikin
\item
Suppose that G is a one-relator group containing non-trivial
elements of finite order and N is a
subgroup of G generated by all elements of finite order.
Is it true that any subgroup of G, which
intersects N trivially, is a free group? One can show that the answer
is affirmative in the cases where G/N
has non-trivial centre or satisfies a non-trivial identity.
\medskip\noindent
D.I. Moldavanskii
\item
If a relatively free group is finitely presented, is it almost nilpotent?
\medskip\noindent
A. Yu. Ol'shanskii
\item
Let G be a non-elementary hyperbolic group and let $G^n$
be the subgroup generated by the n-th
powers of the elements of G.
\noindent a) (M. Gromov). Is it true that $G/G^n$ is infinite for some
$ n=n(G)$?
\noindent b) Is it true that $\cap_{n=1}^{\infty}G^n=1$?
\medskip\noindent
A. Yu. Ol'shanski
\item
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 the element of minimal length in R.
Is it true that s is conjugate to one of the
following elements: $r, r^{-1}, [r,f], [r^{-1},f]$
for some $f \in F$?
\medskip\noindent
V.N. Remeslennikov
\item
Is the conjugacy problem soluable for finitely generated
abelian-by-polycyclic groups?
\medskip\noindent
V.N. Remeslennikov
\item
Is the isomorphism problem solvable
\noindent
a) for finitely generated metabelian groups?
\noindent
b) for finitely generated soluable groups of finite rank?
\medskip\noindent
V.N. Remeslennikov
\item Is the integral group ring of a torsion-free group an integral
domain?
\item
Let F be a non-cyclic free group and let R be a non-cyclic subgroup of F.
Is it true that if [R,R] is a
normal subgroup of F then R is also a normal subgroup of F?
\medskip\noindent
V.E. Shpil'rain
\item Does there exist a finitely presented group such that
its growth function defined relative to some
finite set of generators, is non-recursive?
\medskip\noindent
R.I. Grigorchuk
\item
Is it true that for every $n \geq 2$
and every two epimorphisms $\phi$ and $\psi$
of a free group $F_{2n}$
of rank 2n onto $F_n \times F_n$
there exists an automorphism $\alpha$ of $F_{2n}$
such that $\alpha \phi = \psi$?
\medskip\noindent
R.I. Grigorchuk
\item Let F be a free group on two generators x, y and let
$\phi$ be the automorphism of $F$ defined by
$$x \mapsto y, y \mapsto xy.$$
Let G be a semidirect product of $F/(F"(F')^2)$ and $<\phi$.
Then G is just-non-polycyclic. What is the cohomological dimension
of G over $\Bbb Q$? (It is either 3 or 4.)
\medskip\noindent
P.H. Kropholler
\item Let G have the presentation $<X;R>$ and suppose that
$$H=<X \cup \{t\};R \cup \{w\}>$$
is obtained from G by adding one new generator
and one new defining relator. Can H be the trivial group?
\item
Find out something about the elements of finite
order in the automorphism group of a finitely generated
free group F.
\item Let A and B be non-cyclic, finitely
generated subgroups of a free group F. If $\alpha$ is the rank of A,
$\beta$ the rank of B and $\gamma$ the rank of their intersection C,
is
$$(\gamma - 1) \leq (\alpha -1)(\beta -1)?$$
\item Are one-relator groups coherent?
\item Let the trivial group have a
finite balanced presentation
$<X;R>$, i.e. $\mid X\mid = \mid R \mid$. Can this presentation
be reduced to the trivial presentation by a succession of
transformations of the following kinds:
1. Nielsen transformations of X;
2. Nielsen transformations of R;
3. replacing an element of R by a conjugate;
4. Tietze transformations introducing (or deleting) a new
element x of X together with a new relator r in R defining x?
\item Is the IA-automorphism
group of a finitely generated free group finitely presented?
\item Is a finitely generated free-by-cyclic
group finitely related?
\item
Let $S_g$ be the fundamental group of a compact, connected, orientable
surface of genus $g$. An epimorphism
$$\alpha \ : \ S_g \rightarrow F_g \times F_g$$
{\it factors essentially} through a non-trivial free product
$A \ast B$ if there is a homomorphism
$\beta$ of $S_g$ into $A \ast B$ and a homomorphism
$\gamma$ of $A \ast B$ into $F_g \times F_g$ such that
$\alpha = \beta \gamma$. Then we have the following:
According to the Poincare conjecture, a compact,
connected, simply-connected 3-manifold is homeomorphic to the 3-sphere.
Stallings has proved that the Poincare conjecture is true if and
only if, for every $g>1$ every epimorphism
$$\alpha \ : \ S_g \rightarrow F_g \times F_g$$
factors essentially through a free product.
\item Compute an isoperimetric function for
Thurston's group.
\item Does every non-trivial hyperbolic group have a non-trivial finite
image?
(Use Higman's group as the quotient in the Rip's construction and map into
the fp group containing all finite groups.)
\item Is the group
$$G =<a,b,c;a^{-1}b^2a=b^3,b^{-1}c^2 b=c^3, c^{-1}a^2 c=a^3$$
simple?
(For this we need two routines - adding a relation w=1 and checking to
see if the group is trivial.
And looking for a homomorphism of it into something suitable.)
\end{enumerate}
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