Related papers: The congruence subgroup problem
We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2…
The purpose of this paper is to find an equation with respect to the principal congruence subgroup of level eight without considering modular functions.
Let $G$ be a finite group, $\Z G$ the integral group ring of $G$ and $\U(\Z G)$ the group of units of $\Z G$. The Congruence Subgroup Problem for $\U(\Z G)$ is the problem of deciding if every subgroup of finite index of $\U(\Z G)$ contains…
We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.
In the suborbital graphs studies, there has been a research gap in the sense that the Modular group is connected to two numbers. Thus, this paper attempts to contribute to the studies developed by Gauss, Bolyai, Lobachevsky and Riemann.…
This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection…
In this note we give some new results concerning the subgroup commutativity degree of a finite group $G$. These are obtained by considering the minimum of subgroup commutativity degrees of all sections of $G$.
Congruence families, i.e., $\ell$-adic convergence for well-defined arithmetic subsequences, is a commonplace phenomenon for the coefficients of modular forms. Such families superficially resemble one another, but they often vary…
Groups, in which every subgroup containing some fixed primary cyclic subgroup has a complement, are investigated.
In this paper, we study combinatorics of congruence subgroups of the modular group. More precisely, we consider the matrix equation that naturally arises in the theory of Coxeter friezes and investigate its irreducible solutions. We give…
The symmetries of Unimodular Gravity are clarified somewhat.
The purpose of this note is to discuss several results that have been obtained in the last decade in the context of sharp adjoint Fourier restriction/Strichartz inequalities. Rather than aiming at full generality, we focus on several…
Understanding the convergence process of neural networks is one of the most complex and crucial issues in the field of machine learning. Despite the close association of notable successes in this domain with the convergence of artificial…
We give a computationally effective criterion for determining whether a finite-index subgroup of SL(2, Z) is a congruence subgroup, extending earlier work of Hsu for subgroups of PSL(2, Z).
We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…
The aim of this article is to draw attention towards various natural but unanswered questions related to the lower central series of the unit group of an integral group ring.
This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some…
The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation $B_n \to \operatorname{GL}_{n}(\mathbb Z)$. We find the image of such congruence subgroups in $\operatorname{GL}_{n}(\mathbb…
The paper investigates invariants of compactified Picard modular surfaces by principal congruence subgroups of Picard modular groups. The applications to the surface classification and modular forms are discussed.
This paper enriches the list of known properties of congruence sequences starting from the universal relation and successively performing the operators lower $k$ and lower $t$. Two series of inverse semigroups, namely…