Weak Weyl's Law For 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...

On Cool Congruence Formats for Weak Bisimulations

In TCS 146, Bard Bloom presented rule formats for four main notions of bisimulation with silent moves. He proved that weak bisimulation equivalence is a congruence for any process algebra defined by WB cool rules, and established similar results for rooted weak bisimulation (Milner’s “observational congruence”), branching bisimulation and rooted branching bisimulation. This study reformulates B...

Weak-Minded Responsibility in Iranian Criminal Law and Egyptian Law

In Iran's criminal law, humans are not only divided into wise or insane they are also persons who, on the one hand, do not fall under Article 149 of the Islamic Penal Code in the definition of insane persons and on the other hand, they are not wise, the rational age of these people is lower than their child's physical age that Affected by mental disorder and mental retardation. Article 91 of th...

Congruence Subgroups of Groups Commensurable with PSL

Stability in the homology of congruence subgroups

The homology groups of many natural sequences of groups {Gn}n=1 (e.g. general linear groups, mapping class groups, etc.) stabilize as n→ ∞. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense a...