do semestre: MA 141
I have been working at the department in mathematics, IMECC,
UNICAMP, Brazil since 2001 and in 2009 I was elected to a
interests are mainly algebraic but some lie on the border line with
topology. Most of my work is about discrete groups, pro-p groups and
Discrete groups: I study the geometric homological and
due to R. Bieri, R. Strebel, B. Renz and W. Newmann. My PhD thesis at
UK was mainly devoted to the FPm Conjecture for metabelian groups. This
a conjecture due to R. Bieri that studies the homological type FPm of a
group via the first geometric invariant of the group. The
is still an open problem though many cases have been proved.
is a monoid version of the FPm Conjecture, the Sigma^m-Conjecture for
groups which is strongly related to the first one, though there are no
results stating that one of the conjectures implies the other. The
Conjecture suggests a description of the higher dimensional geometric
of a metabelian group via the first dimensional invaraint. I have
a survey paper on the subject based on a mini-course given at Algebra
August 2002, Cabo Frio, Brazil, which is available at request.
Pro-p groups : My interest in the homological properties of pro-p
was awaken by the definition of invariant for metabelian pro-p groups
King that was suggested in Jeremy's PhD thesis at Cambridge University,
J. King conjectured a pro-p version of the FPm Conjecture and proved
parts of it. The conjecture was settled by me few years ago, though the
counterpart is still open. Recently in a joint work with P.
we defined homological invariants for any pro-p group of homological
FPm, these new invariants have strong homological nature without any
Lie algebras: Homological properties of Lie algebras L can be studies
of the homological types of the universal enveloping algebra U(L) of L.
are some recent results due to R. Bryant and J. Groves that classify
presented metabelian Lie algebras via a new invaraint for such Lie
The invariant has valuation-theoretic nature and bears similarities
the J.King's invariant for metabelian pro-p groups. There is a version
the FPm Conjecture for metabelian Lie algebras I proved in the
extension case. The general case is still open, though what already is
in the Lie metabelian case is more than in the discrete groups
Recently some of the results for Lie algebras have been
for metabelian Hopf algebras.
Homological properties of modules: There is a generalised version of
Conjecture for metabelian groups suggesting a classification of the
generated Q-modules B (Q finitely generated abelian group) that are of
FPm over a group extension of A by Q (A being finitely generated
using the Bieri-Strebel invariants of A and B as Q-modules. The
case can be obtained when B = Z is the trivial module. Some low
cases of this generalised conjecture are known but it is much more
than the classical case.
Locally compact groups: In a Springer Lecture Notes book H. Abels has
all finitely presented S-arithmetic nilpotent-by-abelian groups by
styding locally compact, compactly presentable groups. Following the
of locally compact, compactly presentable groups I have defined
studied topological version of the 2 dimensional homotopical invariant
discrete groups. Possible future corollaries include classification of
finitely presented subgroups in S-arithmetic nilpotent-by-abelian
contain the commutator.
L^2-methods in group theory : I have been recently interested in
and applying L^2-methods in group theory.
The main idea is to embed the group algebra of G with
coefficients into L^2(G) . The advantage of working with L^2(G) is that
is a Hilbert G--module with well defined dimension ( the von Neumann
for its Hilbert G-submodules. Methods from L^2-theory together with
results of Robert Bieri have enabled me to prove a conjecture due to E.
Strasser : for every knot-like group G with a finitely generated
G' the commutator is always free (this is well known for
groups). The main ingredient of the proof is the fact that
Novikov rings associated to discrete non-zero characters are von
finite (i.e. every left inverse is right inverse).
Homological properties of limit groups and pro-p analogues: I have been
interested in subdirect product of limit groups and is some pro-p
groups that share common properties with limit groups, including
Demushkin groups. Recently I have been working with M. Bridson on
the asymptotics of homology groups of subdirect product of limit
groups, this has lead to the calculation of some low dimensional L^2
Betti numebres of subdirect products.
Bredon homological types : this generalizes the classical
homological type FPm and is calculated with respect to some class of
subgroups. More results are known when the class of finite subgroups is
considered. I have been working on this project with B. Nucinkis and C.
Pro-p completions of Poincare duality groups : I have been studing what
sufficient conditions guarantee that the pro-p completion of an
orientable Poincaré duality group is again Poincaré duality (in the
category of pro-p groups) without assuming that the dimension stays the
same, in all the examples known the dimension might decrease but never
increses. In the case of Poincare duality groups of dimension 3 more
resulys hold, as shown in a joint paper with P. Zalesskii.