Congruence Groups Research Articles

Certain congruences on eventually regular semigroups I

A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

On Monoids of Injective Partial Cofinite Selfmaps

Abstract We study the semigroup Icf λ of injective partial cofinite selfmaps of an infinite cardinal λ. We show that Icf λ is a bisimple inverse semigroup and each chain of idempotents in Icf λ is contained in a bicyclic subsemigroup of Icf λ , we describe the Green relations on Icf λ and we prove that every non-trivial congruence on Icf λ is a group congruence. Also, we describe the structure of the quotient semigroup Icf λ /σ, where σ is the least group congruence on Icf λ.

Kontsevich–Zagier integrals for automorphic Green’s functions. I

In the framework of Kontsevich-Zagier periods, we derive integral representations for weight-$k$ automorphic Green's functions invariant under modular transformations in $\varGamma_0(N)$ ($N\in\mathbb Z_{\geq1} $), provided that there are no cusp forms on the respective Hecke congruence groups with an even integer weight $k\geq4$. These Kontsevich-Zagier integral representations for automorphic Green's functions give explicit formulae for certain Eichler-Shimura maps connecting Eichler cohomology to Maa{\ss} cusp forms. We construct integral representations for weight-4 Gross-Zagier renormalized Green's functions (automorphic self-energy) from limit scenarios of the respective Kontsevich-Zagier integrals. We reduce the weight-4 automorphic self-energy on $ X_0(4)(\mathbb C)=\varGamma_0(4)\setminus\mathfrak H^*$ to an explicit form, which supports an algebraicity conjecture of Gross and Zagier.

Spectral correspondences for Maass waveforms on quaternion groups

We prove that in most cases the Jacquet–Langlands correspondence between newforms for Hecke congruence groups and newforms for quaternion orders is a bijection. Our proof covers almost all cases where the Hecke congruence group is of cocompact type, i.e. when a bijection is possible. The proof uses the Selberg trace formula.

The structure of certain F-inverse semigroups

An inverse semigroup is an F-inverse semigroup if each class modulo the minimum group congruence has a maximum element. This paper studies a class of F-inverse semi-groups in which all these maximum elements form a subgroup. After obtaining some characterizations of such semigroup, the structure theorem is obtained by the semidirect product of a semilattice and a group.

Ramanujan series for Epstein zeta functions

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions $$E^{\varGamma _0(N)}(z,s)$$ on the Hecke congruence groups $$ \varGamma _0(N),N\in \mathbb {Z}_{>0}$$ , where z is an arbitrary point in the upper half-plane $$ \mathfrak {H}$$ and $$s\in \mathbb {Z}_{>1}$$ . These Ramanujan series can be reformulated as integrations of modular forms, in the framework of Eichler integrals. Particular cases of these Eichler integrals recover part of the recent results reported by Wan and Zucker ( arXiv:1410.7081v1 , 2014).

Group congruences on Q-regular semigroups

Group congruences on Q-regular semigroups

On the low-dimensional homology of [formula omitted

We prove analogues of the fundamental theorem of K-theory for the second and the third homology of SL2 over an infinite field. The statements of the theorems involve Milnor–Witt K-theory and refined scissors congruence groups. We use these results to calculate the low-dimensional homology of SL2 of Laurent polynomials over certain fields.

Members’ perceptions of person-centered facilitative conditions and their role in outcome in a psychoeducational group for childhood social anxiety

Research in counseling and psychotherapy groups firmly supports the view that a positive member-leader relationship is necessary for client change. However, studies that investigate the ingredients of this relationship are scarce. Forty elementary school children selected for displaying medium to high levels of social anxiety completed several scales measuring emotional traits, cognitions, and social skills, both before and after the termination of the intervention. The Barrett-Lennard Relationship Inventory tapping into members’ perception of facilitative attitudes of a leader (i.e. empathic understanding, congruence, and unconditional positive regard) was also administered twice (at session 3 and after the termination). It was found that a medium change in perception of group leader’s facilitative attitudes from the third to the last session was associated with the greatest reductions in social anxiety and depression scores. Moreover, children who perceived high group leader empathy, congruence, or regard at session 3 reported a significant increase in self-reported likeability, compared to children who evidenced a low perception of the same attitudes. The current findings appear to support the association of leader person-centered attitudes with counseling outcome in a psychoeducational group for social anxiety.

Amenability of semigroups and their algebras modulo a group congruence

We investigate the amenability of the semigroup algebras \({\ell^1(S/\rho)}\) , where \({\rho}\) is a group congruence (not necessarily minimal) on a semigroup S. We relate this to a new notion of amenability of Banach algebras modulo an ideal, to prove a version of Johnson’s theorem for a large class of semigroups, including inverse semigroups, E-inversive semigroup and E-inversive E-semigroups.

Euler products beyond the boundary for Selberg zeta functions

Convergence of Euler products in the critical strip is directly related to a proof of the generalized Riemann hypothesis. Moreover its behavior on the critical line is called the deep Riemann hypothesis (DRH). Kimura-Koyama-Kurokawa recently proved DRH over function fields in case the $L$-function is regular at $s=1$ [3]. In this paper we generalize their results to Selberg zeta functions. Our results imply the DRH for principal congruence groups over function fields.

Lifts of projective congruence groups, II

We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\SL_2(\Z)$ that are projectively equivalent to one of the groups $\Gamma_0(N)$ or $\Gamma_1(N)$. A complete answer to this question is obtained: In case of $\Gamma_0(N)$ such noncongruence subgroups exist precisely if $N\not\in {3,4,8}$ and we additionally have either that $4\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of $\Gamma_1(N)$ these noncongruence subgroups exist precisely if $N>4$. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of $\SL_2(\Z)$ that can serve as basis for experimentation with modular forms on noncongruence subgroups.

Fuzzy group congruences on an E-inversive semigroup

Fuzzy group congruences on an E-inversive semigroup

The Deficiency of being a Congruence Group for Veech Groups of Origamis

We study "how far away" a finite index subgroup G of SL(2,Z) is from being a congruence group. For this we define its deficiency of being a congruence group. We show that the index of the image of G in SL(2,Z/nZ) is biggest, if n is the general Wohlfahrt level. We furthermore show that the Veech groups of origamis (or square-tiled surfaces) in the stratum H(2) are far away from being congruence groups and that in each genus one finds an infinite family of origamis such that they are "as far as possible" from being a congruence group.

It Takes Two: Sensitive Caregiving Across Contexts and Children's Social, Emotional, and Academic Outcomes

Research Findings: Using longitudinal survey data from the Welfare, Children, and Families Study: A Three-City Study (n = 135), this study examines how congruence in maternal and child care provider sensitivities contributes to young children's social, emotional, and academic outcomes among low-income minority families. Congruence groups were created based on levels of high and low maternal and child care provider sensitivity. Children with high maternal sensitivity and low child care provider sensitivity had lower scores on measures of social competence and applied problems compared to children with high maternal and child care provider sensitivity. Children with low maternal sensitivity but high child care provider sensitivity displayed higher emotional competence than children with low maternal and child care sensitivity, implying an important protective benefit of child care. Practice or Policy: Current state and federal policy climates, including recently awarded Early Learning Challenge grants focused on social and personal development and the Academic, Social, and Emotional Learning Act, reflect an important emphasis on social and emotional learning. Given this, the findings from this study implicate the role of families and child care providers as important components in any policy or program focused on shaping children's early social and emotional outcomes.

Representation theory and homological stability

We introduce the idea of representation stability (and several variations) for a sequence of representations Vn of groups Gn. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood–Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1)n−1 conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures.Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied by Church–Ellenberg–Farb (in preparation) [20] to counting problems in number theory and finite group theory. Representation stability is also used by Church (2012) [19] to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.

Completely inverse AG ∗∗-groupoids

A completely inverse $AG^{**}$-groupoid is a groupoid satisfying the identities $(xy)z=(zy)x$, $x(yz)=y(xz)$ and $xx^{-1}=x^{-1}x$, where $x^{-1}$ is a unique inverse of $x$, that is, $x=(xx^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}$. First we study some fundamental properties of such groupoids. Then we determine certain fundamental congruences on a completely inverse $AG^{**}$-groupoid; namely: the maximum idempotent-separating congruence, the least $AG$-group congruence and the least $E$-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse $AG^{**}$-groupoids. In particular, we describe congruences on completely inverse $AG^{**}$-groupoids by their kernel and trace.

Are We on the Same Page? The Performance Effects of Congruence between Supervisor and Group Trust

Tapping both cognitive and affective trust, we examined symmetrical supervisor-work group congruence effects on individual group members’ task and contextual performance using a polynomial regressi...

Rectangular group congruences on a semigroup

We study rectangular group congruences on an arbitrary semigroup. Some of our results are an extension of the results obtained by Masat (Proc. Am. Math. Soc. 50:107–114, 1975). We show that each rectangular group congruence on a semigroup S is the intersection of a group congruence and a matrix congruence and vice versa, and this expression is unique, when S is E-inversive. Finally, we prove that every rectangular group congruence on an E-inversive semigroup is uniquely determined by its kernel and trace.

Congruences and group congruences on a semigroup

We show that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup S and the set of all group congruences on S. We describe also group congruences on E-inversive (E-)semigroups. In particular, we generalize the result of Meakin (J. Aust. Math. Soc. 13:259–266, 1972) concerning the description of the least group congruence on an orthodox semigroup, the result of Howie (Proc. Edinb. Math. Soc. 14:71–79, 1964) concerning the description of ρ∨σ in an inverse semigroup S, where ρ is a congruence and σ is the least group congruence on S, some results of Jones (Semigroup Forum 30:1–16, 1984) and some results contained in the book of Petrich (Inverse Semigroups, 1984). Also, one of the main aims of this paper is to study of group congruences on E-unitary semigroups. In particular, we prove that in any E-inversive semigroup, \(\mathcal{H}\cap\sigma\subseteq\kappa\), where κ is the least E-unitary congruence. This result is equivalent to the statement that in an arbitrary E-unitary E-inversive semigroup S, \(\mathcal{H}\cap\sigma= 1_{S}\).