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Convergence of symmetries: nilpotency, dimension, and perfectness
Abstract
Let (Xi,pi) be a sequence of pointed n-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and Gi ≤ Iso (Xi) a sequence of closed groups of isometries. We show that if the triples (Xi, Gi, pi) converge in the equivariant Gromov--Hausdorff sense to a triple (X,G,p), then g, the Lie algebra of G, admits an ideal h trianglelefteq g with dim (h) ≤ limsupi dim (Gi) and g/ h nilpotent. Moreover, if the sequence $(X_i, p_i ) $ is non-collapsing, we show that $\mathfrak{h} $ can be taken of dimension limsupi dim (Gi).
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BibTeX
@misc{zamora2026convergence,
title = {{Convergence of symmetries: nilpotency, dimension, and perfectness}},
author = {Sergio Zamora},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30990},
eprint = {2608.30990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30990},
}
RIS
TY - UNPB
TI - Convergence of symmetries: nilpotency, dimension, and perfectness
AU - Sergio Zamora
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2608.30990
UR - https://arxiv.org/abs/2608.30990
AB - Let (X_i,p_i) be a sequence of pointed n-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and G_i ≤ Iso (X_i) a sequence of closed groups of isometries. We show that if the triples (X_i, G_i, p_i) converge in the equivariant Gromov--Hausdorff sense to a triple (X,G,p), then g, the Lie algebra of G, admits an ideal h trianglelefteq g with dim (h) ≤ limsup_i dim (G_i) and g/ h nilpotent. Moreover, if the sequence $(X_i, p_i ) $ is non-collapsing, we show that $\mathfrak{h} $ can be taken of dimension limsup_i dim (G_i).
ER -