Unified treatment of Artin-type problems
نویسندگان
چکیده
Abstract Since Hooley’s seminal 1967 resolution of Artin’s primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations have been considered. We present a framework generalizing and unifying many previously considered variants, prove results in this full generality (under GRH).
منابع مشابه
Problems Related to Artin Groups
Artin groups span a wide range of groups from braid groups to free groups to free abelian groups, as well as many more exotic groups. In recent years, Artin groups and their subgroups have proved to be a rich source of examples and counterexamples of interesting phenomena in geometry and group theory. Artin groups, like Coxeter groups, are defined by presentations of a particular form. A Coxete...
متن کاملArtin groups of euclidean type
Coxeter groups were introduced by Jacques Tits in the 1960s as a natural generalization of the groups generated by reflections which act geometrically (which means properly discontinuously cocompactly by isometries) on spheres and euclidean spaces. And ever since their introduction their basic structure has been reasonably well understood [BB05, Bou02, Dav08]. More precisely, every Coxeter grou...
متن کاملLinearity of Artin Groups of Finite Type
Recent results on the linearity of braid groups are extended in two ways. We generalize the Lawrence Krammer representation as well as Krammer’s faithfulness proof for this linear representation to Artin groups of finite type.
متن کاملCovers of D-Type Artin Groups
We study certain quotients of generalized Artin groups which have a natural map onto D-type Artin groups, where the generalized Artin group A(T ) is defined by a signed graph T . Then we find a certain quotient G(T ) according to the graph T , which also have a natural map onto A(Dn). We prove that G(T ) is isomorphic to a semidirect product of a group K(m,n), with the Artin group A(Dn), where ...
متن کاملA Unified Treatment of Some Basic Problems in Homotopy Theory
0. Introduction. This note announces some new methods in algebraic topology, based on the results of [1], [2] and [3]. The relative lifting problem [5, p. 415] is fundamental to that subject. It includes the extension problem, the retraction problem, the lifting problem, the section problem, the relative section problem and the computation of [X, Y] problem, among its particular cases. These pr...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Research in number theory
سال: 2022
ISSN: ['2363-9555', '2522-0160']
DOI: https://doi.org/10.1007/s40993-022-00418-6