Exotic Unit Groups of Sylvester Rank Completions
Abstract
Let R be any unital ring equipped with a Sylvester matrix rank function, and let Q be the rank completion of the matrix system defined by block-diagonal embeddings along a factor sequence. We prove that both GL(Q) and (Q,+) are exotic: every strongly continuous unitary representation is trivial. We also prove that GL(Q) is Bourbaki bounded and admits no nonzero escape function. For every ε>0, let Bε be the open ball centered at the identity. The least positive integer N such that GL(Q)=BεN is ⌊ε−1⌋+1. Neither regularity nor irreducibility is required. Our proof of exoticness uses a uniform Kazhdan estimate for additive integer matrix groups together with rank-one matrix perturbations, rather than the character-theoretic and continuous ring methods used previously. The geometric conclusions follow from factorizations using diagonal corner subgroups.
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BibTeX
@misc{jiang2026exotic,
title = {{Exotic Unit Groups of Sylvester Rank Completions}},
author = {Baojie Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24212},
eprint = {2609.24212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24212},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exotic Unit Groups of Sylvester Rank Completions
AU - Baojie Jiang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24212
UR - https://arxiv.org/abs/2609.24212
AB - Let R be any unital ring equipped with a Sylvester matrix rank function, and let Q be the rank completion of the matrix system defined by block-diagonal embeddings along a factor sequence. We prove that both GL(Q) and (Q,+) are exotic: every strongly continuous unitary representation is trivial. We also prove that GL(Q) is Bourbaki bounded and admits no nonzero escape function. For every ε>0, let B_ε be the open ball centered at the identity. The least positive integer N such that GL(Q)=B_ε^N is ⌊ε^−1⌋+1. Neither regularity nor irreducibility is required. Our proof of exoticness uses a uniform Kazhdan estimate for additive integer matrix groups together with rank-one matrix perturbations, rather than the character-theoretic and continuous ring methods used previously. The geometric conclusions follow from factorizations using diagonal corner subgroups.
ER -