arXiv subjects

Helge Glöckner

Publications and source records attributed to Helge Glöckner.

4 recordsLinked to original sources

arXiv2026

Compact Invariant Random Subgroups

We study ergodic invariant random subgroups that give full measure to the subset of compact subgroups. We show that in real Lie groups, compactly generated $p$-adic Lie groups, locally compact hyperbolic groups and infinitely ended groups they are always contained in a compact normal subgroup. In general $p$-adic Lie groups, we show they are contained in the locally elliptic radical. In totally disconnected locally compact groups, we show they are contained in the intersection of all Levi subgroups of inner automorphisms.

math.GR↗

arXiv2025

Boundary values of diffeomorphisms of simple polytopes, and controllability

We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup of face respecting diffeomorphisms and its Lie theoretic properties. We find a canonical Lie group structure for the quotient of the diffeomorphism by the subgroup Diff$^{\partial,id}(M)$ of maps that equal the identity on the boundary, turning the canonical quotient homomorphism Diff$(M)\rightarrow $Diff$(M)/$Diff$^{\partial,id}(M)$ into a smooth submersion. We also show that the identity component of the diffeomorphism group is generated by the exponential image, by proving general controllability results.

math.GR↗

arXiv2023

Deep neural networks on diffeomorphism groups for optimal shape reparameterization

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.

math.OC↗

arXiv2017

Contraction Groups and Passage to Subgroups and Quotients for Endomorphisms of Totally Disconnected Locally Compact Groups

The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.

math.GR↗