PreprintarXiv licencearXiv
A faithful representation and profinite completion
Abstract
We exhibit two finitely generated residually finite groups G1 and G2 with isomorphic profinite completions G1̂ ≅ G2̂, such that G1 has a faithful representation into GL(d,ℂ) but G2 is not linear. This answers a question of Andrei Jaikin-Zapirain, Alexander Lubotzky, and Martin Bridson negatively. We also present two finitely generated residually finite groups H1 and H2 with isomorphic profinite completions H1̂ ≅ H2̂ satisfying the following. For every field k with characteristic p ≥ 5 and transcendence degree r ≥ 2, there exists d>0 such that H1 has a faithful representation into GL(d,k). However, H2 admits no faithful finite-dimensional representation over any field of positive characteristic.
Related
- Stone profiniteness, reticulation spectra, and profinite-completion obstructions for residuated lattices
- The Burau representation of the braid group is faithful for n = 4
- Regularity of profinite isomorphisms of hyperbolic 3-manifolds
- Polycyclic groups and profinite isomorphism
- Grothendieck Pairs, Functors and Profinite Invariants
BibTeX
@misc{jang2026faithful,
title = {{A faithful representation and profinite completion}},
author = {Wonyong Jang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01672},
eprint = {2611.01672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01672},
}
RIS
TY - UNPB
TI - A faithful representation and profinite completion
AU - Wonyong Jang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01672
UR - https://arxiv.org/abs/2611.01672
AB - We exhibit two finitely generated residually finite groups G_1 and G_2 with isomorphic profinite completions G_1̂ ≅ G_2̂, such that G_1 has a faithful representation into GL(d,ℂ) but G_2 is not linear. This answers a question of Andrei Jaikin-Zapirain, Alexander Lubotzky, and Martin Bridson negatively. We also present two finitely generated residually finite groups H_1 and H_2 with isomorphic profinite completions H_1̂ ≅ H_2̂ satisfying the following. For every field k with characteristic p ≥ 5 and transcendence degree r ≥ 2, there exists d>0 such that H_1 has a faithful representation into GL(d,k). However, H_2 admits no faithful finite-dimensional representation over any field of positive characteristic.
ER -