Disciplina
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 professorship. My research interests are mainly algebraic but some lie on the border line with algebraic topology. Most of my work is about discrete groups, pro-p groups and Lie algebras.

Discrete groups: I study the geometric homological and homotopical invariants due to R. Bieri, R. Strebel, B. Renz and W. Newmann. My PhD thesis at Cambridge, UK was mainly devoted to the FPm Conjecture for metabelian groups. This is a conjecture due to R. Bieri that studies the homological type FPm of a metabelian group via the first geometric invariant of the group. The FPm-Conjecture is still an open problem though many cases have been proved. There is a monoid version of the FPm Conjecture, the Sigma^m-Conjecture for metabelian groups which is strongly related to the first one, though there are no general results stating that one of the conjectures implies the other. The Sigma^m Conjecture suggests a description of the higher dimensional geometric invariants of a metabelian group via the first dimensional invaraint. I have written a survey paper on the subject based on a mini-course given at Algebra School, August 2002, Cabo Frio, Brazil, which is available at request.

Pro-p groups : My interest in the homological properties of pro-p groups was awaken by the definition of invariant for metabelian pro-p groups by J. King that was suggested in Jeremy's PhD thesis at Cambridge University, 1995. J. King conjectured a pro-p version of the FPm Conjecture and proved some parts of it. The conjecture was settled by me few years ago, though the discrete counterpart is still open. Recently in a joint work with P. Zalesskii we defined homological invariants for any pro-p group of homological type FPm, these new invariants have strong homological nature without any geometric flavour.

Lie algebras: Homological properties of Lie algebras L can be studies by means of the homological types of the universal enveloping algebra U(L) of L. There are some recent results due to R. Bryant and J. Groves that classify finitely presented metabelian Lie algebras via a new invaraint for such Lie algebras. The invariant has valuation-theoretic nature and bears similarities with the J.King's invariant for metabelian pro-p groups. There is a version of the FPm Conjecture for metabelian Lie algebras I proved in the split extension case. The general case is still open, though what already is known in the Lie metabelian case is more than in the discrete groups case. Recently some of the results for Lie algebras have been generalised for metabelian Hopf algebras.

Homological properties of modules: There is a generalised version of the FPm Conjecture for metabelian groups suggesting a classification of the finitely generated Q-modules B (Q finitely generated abelian group) that are of type FPm over a group extension of A by Q (A being finitely generated Q-module) using the Bieri-Strebel invariants of A and B as Q-modules. The classical case can be obtained when B = Z is the trivial module. Some low dimensional cases of this generalised conjecture are known but it is much more difficult than the classical case.

Locally compact groups: In a Springer Lecture Notes book H. Abels has classified all finitely presented S-arithmetic nilpotent-by-abelian groups by firstly styding locally compact, compactly presentable groups. Following the study of locally compact, compactly presentable groups I have defined and studied topological version of the 2 dimensional homotopical invariant for discrete groups. Possible future corollaries include classification of the finitely presented subgroups in S-arithmetic nilpotent-by-abelian groups that contain the commutator.

L^2-methods in group theory : I have been recently interested in learning and applying L^2-methods in group theory.

The main idea is to embed the group algebra of G with integral coefficients into L^2(G) . The advantage of working with L^2(G) is that it is a Hilbert G--module with well defined dimension ( the von Neumann dimension) for its Hilbert G-submodules. Methods from L^2-theory together with some results of Robert Bieri have enabled me to prove a conjecture due to E. Rapaport Strasser : for every knot-like group G with a finitely generated commutator G' the commutator is always free (this is well known for knot groups). The main ingredient of the proof is the fact that the Novikov rings associated to discrete non-zero characters are von Neumann 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. Martinez Perez.

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.

I have been working at the department in mathematics, IMECC, UNICAMP, Brazil since 2001 and in 2009 I was elected to a professorship. My research interests are mainly algebraic but some lie on the border line with algebraic topology. Most of my work is about discrete groups, pro-p groups and Lie algebras.

Discrete groups: I study the geometric homological and homotopical invariants due to R. Bieri, R. Strebel, B. Renz and W. Newmann. My PhD thesis at Cambridge, UK was mainly devoted to the FPm Conjecture for metabelian groups. This is a conjecture due to R. Bieri that studies the homological type FPm of a metabelian group via the first geometric invariant of the group. The FPm-Conjecture is still an open problem though many cases have been proved. There is a monoid version of the FPm Conjecture, the Sigma^m-Conjecture for metabelian groups which is strongly related to the first one, though there are no general results stating that one of the conjectures implies the other. The Sigma^m Conjecture suggests a description of the higher dimensional geometric invariants of a metabelian group via the first dimensional invaraint. I have written a survey paper on the subject based on a mini-course given at Algebra School, August 2002, Cabo Frio, Brazil, which is available at request.

Pro-p groups : My interest in the homological properties of pro-p groups was awaken by the definition of invariant for metabelian pro-p groups by J. King that was suggested in Jeremy's PhD thesis at Cambridge University, 1995. J. King conjectured a pro-p version of the FPm Conjecture and proved some parts of it. The conjecture was settled by me few years ago, though the discrete counterpart is still open. Recently in a joint work with P. Zalesskii we defined homological invariants for any pro-p group of homological type FPm, these new invariants have strong homological nature without any geometric flavour.

Lie algebras: Homological properties of Lie algebras L can be studies by means of the homological types of the universal enveloping algebra U(L) of L. There are some recent results due to R. Bryant and J. Groves that classify finitely presented metabelian Lie algebras via a new invaraint for such Lie algebras. The invariant has valuation-theoretic nature and bears similarities with the J.King's invariant for metabelian pro-p groups. There is a version of the FPm Conjecture for metabelian Lie algebras I proved in the split extension case. The general case is still open, though what already is known in the Lie metabelian case is more than in the discrete groups case. Recently some of the results for Lie algebras have been generalised for metabelian Hopf algebras.

Homological properties of modules: There is a generalised version of the FPm Conjecture for metabelian groups suggesting a classification of the finitely generated Q-modules B (Q finitely generated abelian group) that are of type FPm over a group extension of A by Q (A being finitely generated Q-module) using the Bieri-Strebel invariants of A and B as Q-modules. The classical case can be obtained when B = Z is the trivial module. Some low dimensional cases of this generalised conjecture are known but it is much more difficult than the classical case.

Locally compact groups: In a Springer Lecture Notes book H. Abels has classified all finitely presented S-arithmetic nilpotent-by-abelian groups by firstly styding locally compact, compactly presentable groups. Following the study of locally compact, compactly presentable groups I have defined and studied topological version of the 2 dimensional homotopical invariant for discrete groups. Possible future corollaries include classification of the finitely presented subgroups in S-arithmetic nilpotent-by-abelian groups that contain the commutator.

L^2-methods in group theory : I have been recently interested in learning and applying L^2-methods in group theory.

The main idea is to embed the group algebra of G with integral coefficients into L^2(G) . The advantage of working with L^2(G) is that it is a Hilbert G--module with well defined dimension ( the von Neumann dimension) for its Hilbert G-submodules. Methods from L^2-theory together with some results of Robert Bieri have enabled me to prove a conjecture due to E. Rapaport Strasser : for every knot-like group G with a finitely generated commutator G' the commutator is always free (this is well known for knot groups). The main ingredient of the proof is the fact that the Novikov rings associated to discrete non-zero characters are von Neumann 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. Martinez Perez.

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.