11 research outputs found
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
The irrationality of a number theoretical series
Denote by the sum of the -th powers of the divisors of ,
and let . We prove that Schinzel's
conjecture H implies that is irrational, and give an unconditional proof
for the case
The subgroup growth spectrum of virtually free groups
For a finitely generated group denote by the growth
coefficient of , that is, the infimum over all real numbers such
that . We show that the growth coefficient of a virtually
free group is always rational, and that every rational number occurs as growth
coefficient of some virtually free group. Moreover, we describe an algorithm to
compute