scholarly journals The congruence subgroup property for the hyperelliptic modular group: the open surface case

2009 ◽  
Vol 39 (3) ◽  
pp. 351-362 ◽  
Author(s):  
Marco Boggi
Keyword(s):  
Modular Group ◽  
Congruence Subgroup ◽  
Open Surface ◽  
Congruence Subgroup Property

scholarly journals The profinite completion of multi-EGS groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anitha Thillaisundaram ◽  
Jone Uria-Albizuri
Keyword(s):  
Congruence Subgroup ◽  
Profinite Completion ◽  
Congruence Subgroup Property

AbstractThe class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

scholarly journals Algorithms for arithmetic groups with the congruence subgroup property

2015 ◽  
Vol 421 ◽  
pp. 234-259 ◽  
Author(s):  
A.S. Detinko ◽  
D.L. Flannery ◽  
A. Hulpke
Keyword(s):  
Congruence Subgroup ◽  
Arithmetic Groups ◽  
Congruence Subgroup Property

scholarly journals Congruence testing for odd subgroups of the modular group

2014 ◽  
Vol 17 (1) ◽  
pp. 206-208
Author(s):  
Thomas Hamilton ◽  
David Loeffler
Keyword(s):  
Finite Index ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Index Subgroup ◽  
Finite Index Subgroup

AbstractWe give a computationally effective criterion for determining whether a finite-index subgroup of $\mathrm{SL}_2(\mathbf{Z})$ is a congruence subgroup, extending earlier work of Hsu for subgroups of $\mathrm{PSL}_2(\mathbf{Z})$.

scholarly journals On Siegel modular forms part II

1995 ◽  
Vol 138 ◽  
pp. 179-197 ◽  
Author(s):  
Bernhard Runge
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Graded Ring

In this paper we compute dimension formulas for rings of Siegel modular forms of genus g = 3. Let denote the main congruence subgroup of level two, the Hecke subgroup of level two and the full modular group. We give the dimension formulas for genus g = 3 for the above mentioned groups and determine the graded ring of modular forms with respect to .

scholarly journals On the representation of a number as a sum of squares

1978 ◽  
Vol 19 (2) ◽  
pp. 173-197 ◽  
Author(s):  
Karl-Bernhard Gundlach
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Remainder Term ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Hilbert Modular Group ◽  
Totally Positive ◽  
Hilbert Modular ◽  
Image Position ◽  
Totally Real

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the formwhere Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact thatis a modular form for a certain congruence subgroup of the modular group, andwithwhere Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

Valuation-like maps and the congruence subgroup property

2001 ◽  
Vol 144 (3) ◽  
pp. 571-607 ◽  
Author(s):  
Andrei S. Rapinchuk ◽  
Yoav Segev
Keyword(s):  
Congruence Subgroup ◽  
Congruence Subgroup Property

scholarly journals On the lattice of subgroups of the lamplighter group

2014 ◽  
Vol 24 (06) ◽  
pp. 837-877 ◽  
Author(s):  
R. Grigorchuk ◽  
R. Kravchenko
Keyword(s):  
Binary Tree ◽  
Congruence Subgroup ◽  
Profinite Completion ◽  
Scale Invariant ◽  
Lamplighter Group ◽  
Invariant Structures ◽  
Self Similar ◽  
Lattice Subgroup ◽  
Lattice Of Subgroups ◽  
Congruence Subgroup Property

The techniques of modules and actions of groups on rooted trees are applied to study the subgroup structure and the lattice subgroup of lamplighter type groups of the form ℒn,p = (ℤ/pℤ)n ≀ ℤ for n ≥ 1 and p prime. We completely characterize scale invariant structures on ℒ1,2. We determine all points on the boundary of binary tree (on which ℒ1,p naturally acts in a self-similar manner) with trivial stabilizer. We prove the congruence subgroup property (CSP) and as a consequence show that the profinite completion [Formula: see text] of ℒ1,p is a self-similar group generated by finite automaton. We also describe the structure of portraits of elements of ℒ1,p and [Formula: see text] and show that ℒ1,p is not a sofic tree shift group in the terminology of [T. Ceccherini-Silberstein, M. Coornaert, F. Fiorenza and Z. Sunic, Cellular automata between sofic tree shifts, Theor. Comput. Sci.506 (2013) 79–101; A. Penland and Z. Sunic, Sofic tree shifts and self-similar groups, preprint].

Level and index in the modular group

1984 ◽  
Vol 99 (1-2) ◽  
pp. 115-126 ◽  
Author(s):  
W. W. Stothers
Keyword(s):  
Modular Group ◽  
Congruence Subgroup ◽  
Existence Theorems ◽  
Congruence Subgroups ◽  
Similar Action

SynopsisIt is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup. This allows a number of existence theorems about non-congruence subgroups.The level of a subgroup of the modular group can be defined in terms of the action on Q ∪ {∞}. We define a similar action to get information on congruence subgroups. In fact, we get a more powerful result, but this appears to be the most useful version.

scholarly journals Profinite invariants of arithmetic groups

2020 ◽  
Vol 8 ◽  
Author(s):  
Holger Kammeyer ◽  
Steffen Kionke ◽  
Jean Raimbault ◽  
Roman Sauer
Keyword(s):  
Euler Characteristic ◽  
Finite Groups ◽  
Congruence Subgroup ◽  
Arithmetic Groups ◽  
Profinite Completion ◽  
Residually Finite ◽  
Residually Finite Groups ◽  
Congruence Subgroup Property

Abstract We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$ -torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

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