Counting congruence subgroups

Counting Congruence Subgroups

Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence subgroups of Γ of index at most n. We prove that lim n→∞ log Cn(Γ) (log n)2/ log log n = 3−2 √ 2 4 . We also present a very general conjecture giving an asymptotic estimate for Cn(Γ) for general arithmetic groups. The lower bound of the conjecture is proved modulo the generalized Riemann hypothesis for Artin-He...

Counting Finite Index Subgroups

Let F be a finitely generated group. Denote by an(T) (resp.<rn(r)) the number of subgroups of F of index n (resp. of index at most n). This paper deals with the connection between the algebraic structure of the group F and the arithmetic properties of the sequence an(F), n — 1,2,3,..., e.g., the growth of the sequence an(T) ("the subgroup growth") or the properties of the function Cr(s) = X2£Li...

Counting Maximal Arithmetic Subgroups

for absolute constants C6, C7. This theorem (almost) follows from [EV, Theorem 1.1], the only point being to control the dependence of implicit constants on the degree of the number field. We refer to [EV] for further information and for some motivational comments about the method. In the proof C1, C2, . . . will denote certain absolute constants. A.2. Let K be an extension of Q of degree d ≥ 2...

COUNTING DISTINCT FUZZY SUBGROUPS OF SOME RANK-3 ABELIAN GROUPS

In this paper we classify fuzzy subgroups of a rank-3 abelian group $G = mathbb{Z}_{p^n} + mathbb{Z}_p + mathbb{Z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorp...

Congruence Subgroups of Groups Commensurable with PSL