According to my
knowledge, the following questions are unsolved. If I dont specify otherwise,
they come from me (again, according to my best knowledge). Some may be easy,
some hard. Good luck solving them!
1) (Alon Amit) Is it true that if P is a finite
p-group of order n and w is any word then the probability that w is satisfied
is at least 1/n?
By a result of Amit it is true for any finite
group if w uses at most two variables. It is proved by Nikolov and Segal that
for solvable groups the set of probabilities is bounded away from zero (Amit
has settled the nilpotent case). I showed that for non-solvable groups it
accumulates in zero. (Note that we fix the group and let w run through all
words.)
2) (with Balint Virag) What is the length of the
shortest law in the n times iterated wreath product of C(2)?
A law in a group is a nontrivial word which is
satisfied whatever we substitute into it from the group. Conjecturally the
answer is 2^n and the only such law is the power word x^(2^n). The n times
iterated wreath product of C(2) is isomorphic to the Sylow-2 subgroup of
Sym(2^n). One can ask the same question for any family of groups (e.g. Sym(n)
itself): the answer is unknown for most.
3) What is the length of the shortest nontrivial
word which is satisfied in all groups of size 2^n?
Conjecturally it is 2^n, which would imply the
same answer for Question 5. However, opposed to Question 5, there are a great
many candidates for such a law.
4) Does there exist a countable group with
infinite commutator width and trivial pseudocharacter space?
The commutator width of a group G is the minimal k
such that every element of G’ can be obtained as the product of k
commutators. Pseudocharacters are related to the second bounded cohomology.
Since pseudocharacters are bounded on the set of commutators, finite commutator
width implies trivial psudocharacter space. On the other hand, a result of
Bavard implies that if there exists an element g in G’ and a>0 such that
the commutator width of g^n is at least a*n, then G has nontrivial
pseudocharacter space.
A good candidate for such a G would be the
2-generated free d-solvable group; it is amenable so it has trivial
pseudocharacter space while it seems likely that it has infinite commutator
width for d>3. However, this is an old unsolved problem. Another candidate
could be SL(3,k[x]) where k is a countable field whose transcendence degree is
infinite over its prime field: it is known that this has infinite commutator width.
5) Let d(G) denote the minimal number of
generators for G. Is it true that for all n there exists a residually finite
linear group G such that d(G)=n and d(H)=2 for all finite quotient H of G?
(That is, the profinite completion G^ of G can be topologically generated by 2
elements).
The discrete direct product of alternating groups
provides an example for an infinitely generated group. Noskov showed examples
of finitely generated metabelian groups where d(G^)<d(G), however he also
showed that for metabelian groups d(G) is quadratically bounded by d(G^). Wise
showed that there is no bound for residually finite groups.
6) Let Gamma be a countable subgroup
of SL_2(Q_p) that does not contain parabolic elements and let g be a random
element of SL_2(Q_p). Show that the group generated by Gamma and g does not
contain any parabolic elements a.s.
It is enough to show that there exists such a g;
however, this may not be easier, as for rooted trees it only follows from the
probabilistic result.
7) Does a free group F have a
nontrivial pseudocharacter that is invariant under Aut(F)?
Most likely not.
8) Let M be an infinite
metric space. Is it true that two percolations of M are quasi-isometric with
each other a.s.?
Not known even for Z.
9) Does the Baumslag-Pride theorem
hold for pro-p groups?
The theorem says that every group with r relations
and r+2 generators virtually surjects onto a nonabelian free group. There are many
different proofs in the literature e.g. by Gromov and Lackenby.
10) Let G be a closed transitive
subgroup of the automorphism group of a rooted tree. Is it true that every open
subgroup of G contains a fixedpoint-free element?
Transitive and fixedpoint-free are understood with
respect to the boundary of the tree. This is equivalent to a question from
Fried-Jarden. For pro-p groups it is true.