Box Spaces: Geometry of Finite Quotients

2020 ◽  
pp. 328-343
Author(s):  
Ana Khukhro ◽  
Alain Valette
Keyword(s):  
Finite Quotients

A non-cyclic one-relator group all of whose finite quotients are cyclic

1969 ◽  
Vol 10 (3-4) ◽  
pp. 497-498 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Normal Subgroup ◽  
Residually Finite ◽  
Defining Relation ◽  
Finite Quotient ◽  
Finite Quotients ◽  
Image Position

Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).

scholarly journals Triangles of Baumslag–Solitar Groups

2012 ◽  
Vol 64 (2) ◽  
pp. 241-253 ◽  
Author(s):  
Daniel Allcock
Keyword(s):  
Finite Groups ◽  
Derived Length ◽  
Special Cases ◽  
Finite Quotients ◽  
General Conditions

Abstract Our main result is that many triangles of Baumslag–Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag–Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length ≤ 3. We obtain a lot of information about finite quotients, even when we cannot determine developability.

scholarly journals Discrete groups without finite quotients

2018 ◽  
Vol 248 ◽  
pp. 138-142 ◽  
Author(s):  
Tommaso Cremaschi ◽  
Juan Souto
Keyword(s):  
Discrete Groups ◽  
Finite Quotients

scholarly journals Metabelian groups with the same finite quotients

1974 ◽  
Vol 11 (1) ◽  
pp. 115-120 ◽  
Author(s):  
P.F. Pickel
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Isomorphism Classes ◽  
Split Extensions ◽  
Finite Quotients ◽  
Finitely Presented

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. Two groups G and H are said to have the same finite quotients if F(G) = F(H). We construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring. These groups also give an example of infinitely many nonisomorphic split extensions of a fixed finitely presented metabelian group by a fixed finite abelian group, all having the same finite quotients.

scholarly journals Quantifying residual finiteness of arithmetic groups

2012 ◽  
Vol 148 (3) ◽  
pp. 907-920 ◽  
Author(s):  
Khalid Bou-Rabee ◽  
Tasho Kaletha
Keyword(s):  
Chevalley Group ◽  
Arithmetic Groups ◽  
Residual Finiteness ◽  
Arithmetic Subgroup ◽  
Higher Rank ◽  
Finite Quotients

AbstractThe normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

scholarly journals Determining Fuchsian groups by their finite quotients

2016 ◽  
Vol 214 (1) ◽  
pp. 1-41 ◽  
Author(s):  
M. R. Bridson ◽  
M. D. E. Conder ◽  
A. W. Reid
Keyword(s):  
Fuchsian Groups ◽  
Finite Quotients
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