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On unit groups of purely infinite rings
Abstract
We prove that the unit group of any purely infinite, regular, right self-injective ring admits a universal sequence of words (thus has strong distortion), is minimally almost periodic, has finite involution width, is verbally simple and verbally elliptic, and has uncountable uniform strong cofinality. These findings are consequences of more refined structural results, some of which provide concrete cardinal estimates involving the Goodearl--Boyle dimension of the ring. A central ingredient in our results is a comprehensive study of bounded normal generation, in particular with respect to certain classes of involutions. Along the way, we develop some abstract methods and concepts pertaining to geometric group theory.
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BibTeX
@misc{schneider2026unit,
title = {{On unit groups of purely infinite rings}},
author = {Friedrich Martin Schneider},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29712},
eprint = {2610.29712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29712},
}
RIS
TY - UNPB
TI - On unit groups of purely infinite rings
AU - Friedrich Martin Schneider
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29712
UR - https://arxiv.org/abs/2610.29712
AB - We prove that the unit group of any purely infinite, regular, right self-injective ring admits a universal sequence of words (thus has strong distortion), is minimally almost periodic, has finite involution width, is verbally simple and verbally elliptic, and has uncountable uniform strong cofinality. These findings are consequences of more refined structural results, some of which provide concrete cardinal estimates involving the Goodearl--Boyle dimension of the ring. A central ingredient in our results is a comprehensive study of bounded normal generation, in particular with respect to certain classes of involutions. Along the way, we develop some abstract methods and concepts pertaining to geometric group theory.
ER -