Determining Fuchsian groups by their finite quotients

By Martin R. Bridson, Marston D. E. Conder, Alan W. Reid. 2014

66 citations recorded by the source corpus.

Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ≅ Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F) = C(Γ) implies that F ≅ Γ. If Γ1 < PSL(2, C) and Γ2 < G are nonuniform arithmetic lattices, where G is a semisimple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ≅ PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf8382006ae402c256f4ff9e42aded4039117f1c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Gareth Wilkes
2. Henry Wilton
3. Pavel A. Zalesskii
4. Alan W. Reid
5. Martin R. Bridson

In Geometry and Topology

1. Gareth Wilkes
2. Henry Wilton
3. Pavel A. Zalesskii
4. Martin R. Bridson
5. Alan W. Reid
6. D. McReynolds
7. Charles F. Miller

In Mathematical Physics

1. Gareth Wilkes
2. Henry Wilton
3. Pavel A. Zalesskii
4. Martin R. Bridson
5. Alan W. Reid

Citations in the corpus

• Distinguishing geometries using finite quotients (2014, background): “) Bridson, Conder and Reid [5] have answered the corresponding question for Fuchsian groups positively, while Long and Reid [29] have given a positive answer to a related question.”
• Absolute profinite rigidity and hyperbolic geometry (2018, background): “[8], [11], [13], [14], [26], [28], [50] and [51].”
• Profinite rigidity for Seifert fibre spaces (2015, background): “1 of [BCR14]; for a finite subgroup of the fundamental group of a graph of groups must be conjugate into one of the vertex groups, which here are the copies of Z/2.”