A mathematical preprint examines how properness of subgroup actions on homogeneous spaces of invertible upper triangular matrices can be detected using coarse geometry. In ordinary terms, the approach looks at large-scale separation between transformed copies of the subgroups and uses that separation to derive sufficient conditions for properness.
The work focuses on closed subgroups L and H of B(n), the group of invertible upper triangular matrices. Its central setting is the homogeneous space B(n)/H, where L acts naturally, alongside an identified space of the form R^(n-1).
Turning properness into a large-scale test
The key idea is asymptotic disjointness. For closed subgroups of a locally compact Hausdorff group, the paper uses an equivalence saying that this large-scale form of separation, expressed in the LR-coarse structure, matches properness of the natural action on G/H. That gives the proof a geometric target: rather than checking the action point by point, one can study how subgroup images behave far away.
To make that comparison possible, the analysis uses maps from the upper triangular group into Euclidean space and examines the transformed images of L and H. It proves that the relevant maps have the large-scale control and coarse properness needed for this comparison, linking algebraic information in the matrices with the geometry used to test the action.
The first main criterion says that the natural L-action on B(n)/H is proper if, for every radius R at least 0, the intersection of the transformed image of L with the radius-R neighborhood of the transformed image of H remains bounded. This is a sufficient condition, rather than a claim that the same test is necessary for every proper action.
A second route uses a map called phi-Theta. If its restriction to L is proper, the natural L-action on B(n)/H is also proper. The paper further states that, for n at least 2 and a nonempty choice of the index set Theta, coarse properness of this restricted map is enough to ensure properness of the L-action on R^(n-1).
The framework reaches beyond the main group
The results are not confined to the full group B(n). For every closed subgroup G of B(n), the composition of the main coarse map with the inclusion of G into B(n) is coarse. This supplies the structural step needed to extend the criterion to closed subgroups of the upper triangular matrix group.
The paper also records specific coarse behavior for the subgroup N(n). When n is at least 2, the map alpha-0 is coarse, each alpha-i is controlled for the relevant indices, and the combined map is coarse when the index set I contains 0. These statements add another set of large-scale map properties for the subgroup analysis.
Small examples show why the details matter
Constructed examples show that the criteria can distinguish different configurations. In the B(4) example, the L-action on B(4)/H is proper. An explicit B(6) example likewise has a proper natural L-action, according to the paper's calculations.
The B(6) case also contains a warning about sensitivity to the subgroup data: changing the sign of the (1, 5)-entry produces a nonproper L-prime action on B(6)/H. The contrast suggests why a test based on transformed subgroup images must retain detailed algebraic information rather than rely only on the broad shape of the matrices.
A separate example in dimension 3 makes the ambient group important. The explicit action is proper on N(3)/H but becomes nonproper when the subgroups are viewed inside B(3). The result shows that properness can depend on which upper triangular group supplies the surrounding geometry.
A useful criterion, with a defined reach
The preprint presents coarse geometry, controlled maps and asymptotic disjointness as an effective framework for deriving sufficient criteria in this upper triangular setting. Its conclusions apply to B(n) and selected closed subgroups, including N(n), rather than to arbitrary locally compact groups or arbitrary homogeneous spaces.
The results are framed as sufficient conditions, not as a claim that these tests are necessary in every case. The examples are specific constructed cases, so they do not indicate how often the criteria would succeed across a broader range of subgroup configurations. The work offers a toolkit for proving properness in specified mathematical settings, not an empirical estimate.
The document is an arXiv preprint, version 1, dated 26 August 2026. It is supported by JST SPRING under Grant Number JPMJSP2132. The analysis leaves open whether conditions of this kind can also be necessary and how far analogous tests could extend beyond subgroups of B(n).
Paper data and sources
Original title: Proper Actions on Homogeneous Spaces of the Upper Triangular Matrix Group via Coarse Geometry
Authors: Hiroaki Nagaya
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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