arXiv subjects

Thomas Weigel

Publications and source records attributed to Thomas Weigel.

At least 19 recordsLinked to original sources

arXiv2026

WiFlow: Estimating Optical Flow using WiFi Channel State Information

Knowing where and how fast objects are moving within a scene is important across various domains. Usually, cameras are used to capture the data necessary for this task, but adding cameras often raises privacy concerns, and the quality of captured frames is heavily influenced by lighting conditions. In this work, we explore using WiFi channel state information (CSI) instead of camera frames for optical flow estimation. We propose WiFlow, a CSI based flow estimator, a preprocessor evaluation for CSI, and three model architectures that offer different trade-offs between accuracy and complexity. Further, we create the first dataset for training and evaluating CSI-based optical flow estimators, and our experiments provide insights into key design elements for this task. Code and data are available at https://visinf.github.io/wiflow.

cs.CV↗

arXiv2026

Generalising a Theorem of Lichtman

We show that under a suitable additional hypothesis the restricted Zassenhaus $\F_p$-Lie algebra or the rational Magnus Lie algebra of a free amalgamated product is the free amalgamated product of the corresponding Lie algebras of the factors. This generalises a Theorem of A.I.\,Lichtman, who proved the analoguous statement for free products. Our conditions include the case when the amalgamated product is a retract in both factors. As a by-product, we show that a free product of residually torsion free nilpotent groups amalgamated along retracts is also residually torsion free nilpotent and obtain also some results on cohomological completeness. In the final sections we apply our main results to two recently raised open questions.

math.GR↗

arXiv2025

Presenting the Zassenhaus Lie algebra by the Magnus Lie algebra

It is shown that the Zassenhaus restricted $\mathbb F_p$-Lie algebra of a (pro-p) group G can be presented by the Magnus Lie algebra of G. For the class of (pro-p) groups for which the terms of the lower central series are torsion-free, the Zassenhaus restricted $\mathbb F_p$-Lie algebra can be explicitly computed from the Magnus Lie algebra. These results apply to orientable surface groups, right-angled Artin groups, pure braid groups, fundamental groups of supersolvable hyperplane arrangements and fundamental groups of strictly supersolvable toric arrangements.

math.GR↗

arXiv2025

On probabilistic identities and coset identities in pro-$p$ groups

It is shown that a probabilistic identity on a $\sigma$-compact $K$-analytic group $G$, $K$ a non-archimedean local field, is a coset identity. As an application, one concludes that compact $K$-analytic groups and various pro-$p$ groups obtained from free constructions satisfy a probabilistic Tits alternative. By means of Lie-theoretic methods, we also study torsion probabilistic identities in virtually free pro-$p$ and compact $p$-adic analytic groups.

math.GR↗

arXiv2025

On commensurators of free groups and free pro-p groups

We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple. For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple.

math.GR↗

arXiv2024

Some invariants of totally disconnected locally compact groups: cohomology and combinatorics

The paper investigates two invariants for totally disconnected locally compact groups: the number of ends and the rational discrete cohomological dimension. For such a compactly generated group $G$ it is shown that its number of ends can be expressed in terms of the rational discrete cohomology of $G$. If $G$ is suitably acting on a building the number of ends and the rational cohomological dimension of $G$ are related to those of the Weyl group associated to the building. In special cases, we are also able to compare the rational discrete cohomological dimension of $G$ to the flat-rank of $G$. Moreover, examples of groups for which these two invariants coincide are given. Our approach leverages the combinatorics of Coxeter groups, yielding new results of independent interest in Coxeter theory. Finally, in the class of totally disconnected locally compact groups acting properly and cocompactly on locally finite buildings, an accessibility result is proved: we explicitly construct a cocompact proper action on a tree if the rational discrete cohomological dimension is one.

math.GR↗

arXiv2024

The Hattori-Stallings rank, the Euler-Poincar\'e characteristic and zeta functions of totally disconnected locally compact groups

For a unimodular totally disconnected locally compact group $G$ we introduce and study an analogue of the Hattori-Stallings rank $\tilde{\rho}(P)\in\mathbf{h}_G$ for a finitely generated projective rational discrete left $\mathbb Q[G]$-module $P$. Here $\mathbf{h}_G$ denotes the $\mathbb Q$-vector space of left invariant Haar measures of $G$. Indeed, an analogue of Kaplansky's theorem holds in this context (cf. Theorem A). As in the discrete case, using this rank function it is possible to define a rational discrete Euler-Poincar\'e characteristic $\tilde{\chi}_G$ whenever $G$ is a unimodular totally disconnected locally compact group of type $\mathrm{FP}_\infty$ of finite rational discrete cohomological dimension. E.g., when $G$ is a discrete group of type $\mathrm{FP}$, then $\tilde{\chi}_G$ coincides with the ''classical'' Euler-Poincar\'e characteristic times the counting measure $\mu_{\{1\}}$. For a profinite group $\mathcal{O}$, $\tilde{\chi}_{\mathcal{O}}$ equals the probability Haar measure $\mu_{\mathcal{O}}$ on $\mathcal{O}$. Many more examples are calculated explicitly (cf. Example 1.7 and Section 5). In the last section, for a totally disconnected locally compact group $G$ satisfying an additional finiteness condition, we introduce and study a formal Dirichlet series $\zeta_{_{G,\mathcal{O}}}(s)$ for any compact open subgroup $\mathcal{O}$. In several cases it happens that $\zeta_{_{G,\mathcal{O}}}(s)$ defines a meromorphic function $\tilde{\zeta}_{_{G,\mathcal{O}}}\colon \mathbb{C} \to\bar{\mathbb C}$ of the complex plane satisfying miraculously the identity $\tilde{\chi}_G=\tilde{\zeta}_{_{G,\mathcal{O}}}(-1)^{-1}\cdot\mu_{\mathcal{O}}$. Here $\mu_{\mathcal{O}}$ denotes the Haar measure of $G$ satisfying $\mu_{\mathcal{O}}(\mathcal{O})=1$.

math.GR↗

arXiv2023

A gentle introduction to Drinfel'd associators

In this note we give an introduction to Drinfel'd's associator coming from the Knizhnik-Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential geometry: we rather use limits of concrete parallel transports.

math.QA↗

arXiv2022

Normal $2$-coverings of the finite simple groups and their generalizations

Given a finite group $G$, we say that $G$ has weak normal covering number $\gamma_w(G)$ if $\gamma_w(G)$ is the smallest integer with $G$ admitting proper subgroups $H_1,\ldots,H_{\gamma_w(G)}$ such that each element of $G$ has a conjugate in $H_i$, for some $i\in \{1,\ldots,\gamma_w(G)\}$, via an element in the automorphism group of $G$. We prove that the weak normal covering number of every non-abelian simple group is at least $2$ and we classify the non-abelian simple groups attaining $2$. As an application, we classify the non-abelian simple groups having normal covering number $2$. We also show that the weak normal covering number of an almost simple group is at least two up to one exception. We determine the weak normal covering number and the normal covering number of the almost simple groups having socle a sporadic simple group. Using similar methods we find the clique number of the invariably generating graph of the almost simple groups having socle a sporadic simple group.

math.GR↗

arXiv2022

On the second cohomology of the norm one group of a p-adic division algebra

Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb Q_p$. Let $D$ be a finite-dimensional central division algebra over $F$ and let $SL_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and Raghunathan proved that $H^2(SL_1(D),\mathbb R/\mathbb Z)$ is a cyclic $p$-group whose order is bounded from below by the number of $p$-power roots of unity in $F$, unless $D$ is a quaternion algebra over $\mathbb Q_2$. In this paper we give an explicit upper bound for the order of $H^2(SL_1(D),\mathbb R/\mathbb Z)$ for $p\geq 5$ and determine $H^2(SL_1(D),\mathbb R/\mathbb Z)$ precisely when $F$ is cyclotomic, $p\geq 19$ and the degree of $D$ is not a power of $p$.

math.GR↗

arXiv2022

Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one

It is shown that a Stallings--Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Thm. B). More precisely, a compactly generated $\mathcal{CO}$-bounded t.d.l.c. group $G$ of rational discrete cohomological dimension less than or equal to $1$ must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody's rational version of the classical Stallings--Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension $1$ has necessarily non-positive Euler--Poincar\'e characteristic (cf. Thm. H).

math.GR↗

arXiv2021

Bass-Serre theory for groupoids

In this paper a Bass-Serre theory in the groupoid setting is developed and a structure theorem is established. Any groupoid action without inversion of edges on a forest induces a graph of groupoids, while any graph of groupoids satisfying certain hypothesis admits a canonical associated groupoid, called the fundamental groupoid, and a forest, called the Bass-Serre forest, such that the fundamental groupoid acts on the Bass-Serre forest. The structure theorem states that these processes are mutually inverse.

math.GR↗

arXiv2020

Profinite groups with a cyclotomic $p$-orientation

Profinite groups with a cyclotomic $p$-orientation are introduced and studied. The special interest in this class of groups arises from the fact that any absolute Galois group $G_{K}$ of a field $K$ is indeed a profinite group with a cyclotomic $p$-orientation $θ_{K,p}\colon G_{K}\to\mathbb{Z}_p^\times$ which is even Bloch-Kato. The same is true for its maximal pro-$p$ quotient $G_{K}(p)$ provided the field $K$ contains a primitive $p^{th}$-root of unity. The class of cyclotomically $p$-oriented profinite groups (resp. pro-$p$ groups) which are Bloch-Kato is closed with respect to inverse limits, free product and certain fibre products. For profinite groups with a cyclotomic $p$-orientation the classical Artin-Schreier theorem holds. Moreover, Bloch-Kato pro-$p$ groups with a cyclotomic orientation satisfy a strong form of Tits' alternative, and the elementary type conjecture formulated by I. Efrat can be restated that the only finitely generated indecomposable torsion free Bloch-Kato pro-$p$ groups with a cyclotomic orientation should be Poincaré duality pro-$p$ groups of dimension less or equal to $2$.

math.GR↗

arXiv2019

Monoids, their boundaries, fractals and $C^\ast$-algebras

In this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]) and the theory of boundary quotients of $C^\ast$-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar M-fractals for a given monoid M, gives rise to examples of $C^\ast$- algebras generalizing the boundary quotients discussed by X. Li in [4, §7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries naturally two topologies. The fine topology plays a prominent role in the construction of these boundary quotients. On the other hand, the cone topology can be used to define canonical measures on the attractor of an M-fractal provided M is finitely 1-generated.

math.AT↗

arXiv2017

Geometric calibration of Colour and Stereo Surface Imaging System of ESA's Trace Gas Orbiter

There are many geometric calibration methods for "standard" cameras. These methods, however, cannot be used for the calibration of telescopes with large focal lengths and complex off-axis optics. Moreover, specialized calibration methods for the telescopes are scarce in literature. We describe the calibration method that we developed for the Colour and Stereo Surface Imaging System (CaSSIS) telescope, on board of the ExoMars Trace Gas Orbiter (TGO). Although our method is described in the context of CaSSIS, with camera-specific experiments, it is general and can be applied to other telescopes. We further encourage re-use of the proposed method by making our calibration code and data available on-line.

astro-ph.IM↗

arXiv2017

Rational discrete cohomology for totally disconnected locally compact groups

Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin's group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$ of type $\mathrm{FP}$ it is possible to define an Euler-Poincaré characteristic $χ(G)$ which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field $K$ and some other examples.

math.GR↗

arXiv2016

Virtually free pro-p products

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).

math.GR↗

arXiv2015

The projective indecomposable modules for the restricted Zassenhaus algebras in characteristic 2

It is shown that for the restricted Zassenhaus algebra $\mathfrak{W}=\mathfrak{W}(1,n)$, $n>1$, defined over an algebraically closed field $\mathbb{F}$ of characteristic 2 any projective indecomposable restricted $\mathfrak{W}$-module has maximal possible dimension $2^{2^n-1}$, and thus is isomorphic to some induced module $\mathrm{ind}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(μ))$ for some torus of maximal dimension $\mathfrak{t}$. This phenomenon is in contrast to the behavior of finite-dimensional simple restricted Lie algebras in characteristic $p>3$.

math.RA↗