Characterizing linear groups in terms of growth properties

Representation Growth of Linear Groups

Let Γ be a group and rn(Γ) the number of its n-dimensional irreducible complex representations. We define and study the associated representation zeta function ZΓ(s) = ∞ ∑ n=1 rn(Γ)n . When Γ is an arithmetic group satisfying the congruence subgroup property then ZΓ(s) has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational rep...

The Growth of Linear Groups

Let G be a group generated by a finite subset S; define S to be the set Ž . < n < of all products of at most n elements of S, and let a S s S be the n n Ž . Ž . Ž . Ž . number of elements in S . As a S satisfies 1 F a S F a S ? a S , n nqm n m Ž .1r n Ž . Ž .1r n the limit lim a S exists, and a S s lim a S G 1. Although the n n Ž . exact value of a S depends on the generating set S, it is well ...

Characterizing Physicochemical Properties of Enset Starch

Characterizing the Multiplicative Group of a Real Closed Field in Terms of its Divisible Maximal Subgroup

Linear Groups and Square Properties in Rings

In [1] a proof was given of Fermat’s Two-Square Theorem using the group theoretical structure of the classical modular group. This has been extended in many directions and to other square properties in general rings. In particular in [2] a two-square theorem was given for the Gaussian integers in terms of when ii is a quadratic residue. In this note we examine and survey this technique and the ...