FREE ABELIAN QUOTIENTS OF COMMENSURATORS
Abstract
For every group Γ, we define a homomorphism d Γ : Comm(Γ) → Z (F S) from the abstract commensurator Comm(Γ) to the free abelian group Z (F S) with basis the collection FS of isomorphism classes of finite simple groups. We investigate the homomorphism d Γ when Γ is a finitely generated free group F . We explicitly describe the image of d F : Comm(F ) → Z (F S) , which is a free abelian group of infinite rank, and we show that the kernel is the monolith of the group Comm(F ). We deduce in particular that every proper quotient of Comm(F ) is abelian.
We use this to study the commensurator of a cocompact lattice in the automorphism group of a regular tree, which can be seen as a subgroup of Comm(F ). We show that the image of this group under d F is again a free abelian group of infinite rank, showing in particular this group is not virtually simple.
Related
- On commensurators of free groups and free pro-p groups
- Abelian quotients and orbit sizes for subgroups of linear groups
- Three Problems on Separable Quotients of Precompact Abelian Groups
- Abelian quotients and the largest orbit sizes of linear groups of odd order
- Descent of Basic Forms to Quotients by Locally Free Lie Group Actions
BibTeX
@misc{boudec2026free,
title = {{FREE ABELIAN QUOTIENTS OF COMMENSURATORS}},
author = {Adrien Le Boudec},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35323},
eprint = {2609.35323},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05768282v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - FREE ABELIAN QUOTIENTS OF COMMENSURATORS
AU - Adrien Le Boudec
PY - 2026
DA - 2026/09/28
PB - HAL
DO - 10.48550/arxiv.2609.35323
UR - https://hal.science/hal-05768282v1
AB - For every group Γ, we define a homomorphism d Γ : Comm(Γ) → Z (F S) from the abstract commensurator Comm(Γ) to the free abelian group Z (F S) with basis the collection FS of isomorphism classes of finite simple groups. We investigate the homomorphism d Γ when Γ is a finitely generated free group F . We explicitly describe the image of d F : Comm(F ) → Z (F S) , which is a free abelian group of infinite rank, and we show that the kernel is the monolith of the group Comm(F ). We deduce in particular that every proper quotient of Comm(F ) is abelian. We use this to study the commensurator of a cocompact lattice in the automorphism group of a regular tree, which can be seen as a subgroup of Comm(F ). We show that the image of this group under d F is again a free abelian group of infinite rank, showing in particular this group is not virtually simple.
ER -