14 research outputs found
On -deficiency in groups
Recently, Schlage-Puchta proved super multiplicity of -deficiency for
normal subgroups of -power index. We extend this result to all normal
subgroups of finite index. We then use the methods of the proof to show that
some groups with non-positive -deficiency have virtually positive
-deficiency. We also compute the -deficiency in some cases such as
Fuchsian groups and study related invariants: the lower and upper absolute
-homology gradients and the -Euler characteristic.Comment: The abstract and introduction were comletely rewritten. Further minor
changes were made for the rest of the pape
Hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum
We prove that the inverse limit of certain iterated wreath products in
product action have complete Hausdorff dimension spectrum with respect to their
unique maximal filtration of open normal subgroups. Moreover we can produce
explicitly subgroups with a specified Hausdorff dimension.Comment: 8 pages, includes minor correction
Large normal subgroup growth and large characteristic subgroup growth
The maximal normal subgroup growth type of a finitely generated group is
. Very little is known about groups with this type of growth. In
particular, the following is a long standing problem: Let be a group
and a subgroup of finite index. Suppose has normal subgroup
growth of type , does has normal subgroup growth of type
? We give a positive answer in some cases, generalizing a result of
M\"uller and the second author and a result of Gerdau. For instance, suppose
is a profinite group and an open subgroup of . We show that if
is a generalized Golod-Shafarevich group, then has normal subgroup growth
of type of . We also use our methods to show that one can find a
group with characteristic subgroup growth of type
Maximal graded subalgebras of loop toroidal Lie algebras, Algebras Represent. Theory
Abstract Let g be a central simple Lie algebra over a field F. We study the maximal Z
Lie algebras with few centralizer dimensions
AbstractIt is known that a finite group with just two different sizes of conjugacy classes must be nilpotent and it has recently been shown that its nilpotence class is at most 3. In this paper we study the analogs of these results for Lie algebras and some related questions
Filtrations in semisimple lie Algebras, II
In this paper, we continue our study of the maximal bounded
Z-filtrations of a complex semisimple Lie algebra L. Specifically, we discuss the functionals which give rise to such filtrations, and we show that they are related to certain semisimple subalgebras of L of full rank. In this way, we determine the “order” of these functionals and count them without the aid of computer computations. The main results here involve the Lie algebras of type E6, E7 and E8, since we already know a good deal about the functionals for the remaining types. Nevertheless, we reinterpret our previous results into the new context considered here. Finally, we describe the associated graded Lie algebras of all of the maximal filtrations obtained in this manner
Abstract commensurators of profinite groups
In this paper we initiate a systematic study of the abstract commensurators
of profinite groups. The abstract commensurator of a profinite group is a
group which depends only on the commensurability class of . We
study various properties of ; in particular, we find two natural ways
to turn it into a topological group. We also use to study topological
groups which contain as an open subgroup (all such groups are totally
disconnected and locally compact). For instance, we construct a topologically
simple group which contains the pro-2 completion of the Grigorchuk group as an
open subgroup. On the other hand, we show that some profinite groups cannot be
embedded as open subgroups of compactly generated topologically simple groups.
Several celebrated rigidity theorems, like Pink's analogue of Mostow's strong
rigidity theorem for simple algebraic groups defined over local fields and the
Neukirch-Uchida theorem, can be reformulated as structure theorems for the
commensurators of certain profinite groups.Comment: 37 pages, final versio