We prove that, if R is a non-discrete irreducible, continuous ring, then its unit group GL(R), equipped with the topology generated by the rank metric, is topologically simple modulo its center, path-connected, locally path-connected, bounded in the sense of Bourbaki, and not admitting any non-zero escape function. All these topological insights are consequences of more refined geometric results concerning the rank metric, in particular with regard to the set of algebraic elements. Thanks to the phenomenon of automatic continuity, our results also have non-trivial ramifications for the underlying abstract groups. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
We present a new algorithm for recovering paths from their third-order signature tensors, an inverse problem in rough analysis. Our algorithm provides the exact solution to this recovery problem and improves upon current approaches by an order of magnitude. It relies on generalized normal forms and stabilizers of group actions via matrix-tensor congruence. We apply randomized transformation techniques that avoid the task of solving nonlinear polynomial systems associated to degenerate paths, and accompany our methods with an efficient implementation in the computer algebra system OSCAR.
Let M be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We call an embedding f : A → M group-extensive if each automorphism of its image extends to an automorphism of M, where the extension map respects composition. We say that M has group-extensible ω-age if each substructure admits a group-extensive embedding into M. We investigate the relationship between the following two properties: the presence of a stationary weak independence relation (SWIR) on M, and group-extensibility of the ω-age of M. We show that linearly ordered Fraïssé structures with a SWIR have group-extensible ω-age, but also we give examples of Fraïssé structures where only one of the two properties holds. Finally, we consider whether a wide range of examples of Fraïssé structures have group-extensible ω-age or a finite SWIR expansion, including all countably infinite ultrahomogeneous oriented graphs (with one exception).
Bass and Kulkarni proved that any pair of free uniform lattices in the automorphism group of a tree have conjugates that both lie inside a third uniform lattice (which is not necessarily free). We show that this does not generalise to trees with fins. The construction of our counter-example involves working with a certain generalisation of the universal groups of Burger and Mozes.
We prove that for almost all symmetric spaces X and for any sequence of compact locally symmetric spaces Y_n which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini–Schramm to X, the joint eigenfunctions of all invariant differential operators on Y_n delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.
