On commensurators of free groups and free pro-p groups

Abstract

We study the commensurators of free groups and free pro-p groups, as well as certain subgroups of these. We prove that the commensurator Comm(F ) of a non-abelian free group of finite rank F is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of Comm(F ) and show that some groups in this family are simple. For a prime p, we also consider the p-commensurator Commp(F ), which is the commensurator of F viewed as a group with pro-p topology. By contrast with Comm(F ), we prove that Commp(F ) has a simple subgroup of index at most 2. Further, while the isomorphism class of Comm(F ) does not depend on the rank of F , we prove that the isomorphism class of Commp(F ) depends on the rank of F and determine the exact dependency. If F is the pro-p completion of F (which is a free pro-p group), Comm(F) is a totally disconnected locally compact (tdlc) group containing F as an open subgroup. We use Commp(F ) to construct an abstractly simple subgroup of Comm(F) containing F as well as a family of non-discrete tdlc groups which are compactly generated and simple.

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BibTeX RIS

BibTeX

@misc{barnea2026commensurators,
  title = {{On commensurators of free groups and free pro-p groups}},
  author = {Yiftach Barnea and Mikhail Ershov and Adrien Le Boudec and Colin D. Reid and Matteo Vannacci and Thomas Weigel},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  eprint = {2507.04120},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05768271v1},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On commensurators of free groups and free pro-p groups
AU  - Yiftach Barnea
AU  - Mikhail Ershov
AU  - Adrien Le Boudec
AU  - Colin D. Reid
AU  - Matteo Vannacci
AU  - Thomas Weigel
PY  - 2026
DA  - 2026/09/22
PB  - HAL
UR  - https://hal.science/hal-05768271v1
AB  - We study the commensurators of free groups and free pro-p groups, as well as certain subgroups of these. We prove that the commensurator Comm(F ) of a non-abelian free group of finite rank F is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of Comm(F ) and show that some groups in this family are simple. For a prime p, we also consider the p-commensurator Commp(F ), which is the commensurator of F viewed as a group with pro-p topology. By contrast with Comm(F ), we prove that Commp(F ) has a simple subgroup of index at most 2. Further, while the isomorphism class of Comm(F ) does not depend on the rank of F , we prove that the isomorphism class of Commp(F ) depends on the rank of F and determine the exact dependency. If F is the pro-p completion of F (which is a free pro-p group), Comm(F) is a totally disconnected locally compact (tdlc) group containing F as an open subgroup. We use Commp(F ) to construct an abstractly simple subgroup of Comm(F) containing F as well as a family of non-discrete tdlc groups which are compactly generated and simple.
ER  -

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