A PROBABILISTIC ZETA FUNCTION FOR ARITHMETIC GROUPS

A profinite group G is positively finitely generated (PFG) if for some k, the probability P(G,k) that k random elements generate G is positive. It was conjectured that if G is PFG, then the function P(G,k) can be interpolated to an analytic function defined in some right half-plane. Here that conjecture is formulated more precisely, and verified for (the profinite completion of) arithmetic groups satisfying the congruence subgroup property.

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Profinite invariants of arithmetic groups

Forum of Mathematics Sigma ◽

10.1017/fms.2020.43 ◽

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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The profinite completion of multi-EGS groups

Journal of Group Theory ◽

10.1515/jgth-2019-0155 ◽

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.

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Algorithms for arithmetic groups with the congruence subgroup property

Journal of Algebra ◽

10.1016/j.jalgebra.2014.08.027 ◽

2015 ◽

Vol 421 ◽

pp. 234-259 ◽

Cited By ~ 8

Author(s):

A.S. Detinko ◽

D.L. Flannery ◽

A. Hulpke

Keyword(s):

Congruence Subgroup ◽

Arithmetic Groups ◽

Congruence Subgroup Property

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On the lattice of subgroups of the lamplighter group

International Journal of Algebra and Computation ◽

10.1142/s0218196714500374 ◽

2014 ◽

Vol 24 (06) ◽

pp. 837-877 ◽

Cited By ~ 11

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].

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Profinite groups in which the probabilistic zeta function has no negative coefficients

International Journal of Algebra and Computation ◽

10.1142/s0218196721500120 ◽

2020 ◽

pp. 1-15

Author(s):

Eloisa Detomi ◽

Andrea Lucchini

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Profinite Group ◽

Möbius Function ◽

Profinite Groups ◽

Finitely Generated ◽

Finitely Generated Group ◽

Mobius Function ◽

Formal Inverse

To a finitely generated profinite group [Formula: see text], a formal Dirichlet series [Formula: see text] is associated, where [Formula: see text] and [Formula: see text] denotes the Möbius function of the lattice of open subgroups of [Formula: see text] Its formal inverse [Formula: see text] is the probabilistic zeta function of [Formula: see text]. When [Formula: see text] is prosoluble, every coefficient of [Formula: see text] is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.

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