scholarly journals Simple Connectivity in Polar Spaces with Group-Theoretic Applications

2014 ◽  
Author(s):  
Reed Nessler
Keyword(s):  
Polar Spaces ◽  
Simple Connectivity

scholarly journals Simple connectivity in polar spaces

2016 ◽  
Vol 28 (3) ◽  
Author(s):  
Max Horn ◽  
Reed Nessler ◽  
Hendrik Van Maldeghem
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Simple Connectivity ◽  
Small Parameters

AbstractWe settle the simple connectivity of the geometry opposite a chamber in a polar space of rank 3 by completing the job for the non-embeddable polar spaces and some polar spaces with small parameters.

On the embeddability of polar spaces

1992 ◽  
Vol 44 (3) ◽  
Author(s):  
Hans Cuypers ◽  
Peter Johnson ◽  
Antonio Pasini
Keyword(s):  
Polar Spaces

scholarly journals Minimal scattered sets and polarized embeddings of dual polar spaces

2007 ◽  
Vol 28 (7) ◽  
pp. 1890-1909 ◽  
Author(s):  
Bart De Bruyn ◽  
Antonio Pasini
Keyword(s):  
Polar Spaces ◽  
Dual Polar Spaces

scholarly journals Opposite Relation on Dual Polar Spaces and Half-Spin Grassmann Spaces

2009 ◽  
Vol 54 (3-4) ◽  
pp. 301-308 ◽  
Author(s):  
Mariusz Kwiatkowski ◽  
Mark Pankov
Keyword(s):  
Polar Spaces ◽  
Dual Polar Spaces

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals On the asymptotic topology of groups and spaces. Part II

2014 ◽  
Vol 55 ◽  
Author(s):  
Daniele Ettore Otera
Keyword(s):  
Simple Connectivity ◽  
Recent Developments

This note continues the study of the topology at infinity of groups and manifolds. Here we will quickly review the notion of geometric simple connectivity together with some more recent developments of it for groups.

scholarly journals Non-Classical Hyperplanes of Finite Thick Dual Polar Spaces

10.37236/7348 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Complete Classification ◽  
Polar Spaces ◽  
Dual Polar Spaces

We obtain a classification of the nonclassical hyperplanes of all finite thick dual polar spaces of rank at least 3 under the assumption that there are no ovoidal and semi-singular hex intersections. In view of the absence of known examples of ovoids and semi-singular hyperplanes in finite thick dual polar spaces of rank 3, the condition on the nonexistence of these hex intersections can be regarded as not very restrictive. As a corollary, we also obtain a classification of the nonclassical hyperplanes of $DW(2n-1,q)$, $q$ even. In particular, we obtain a complete classification of all nonclassical hyperplanes of $DW(2n-1,q)$ if $q \in \{ 8,32 \}$.

scholarly journals Commensurated subgroups, semistability and simple connectivity at infinity

2014 ◽  
Vol 14 (6) ◽  
pp. 3509-3532 ◽  
Author(s):  
Gregory R Conner ◽  
Michael L Mihalik
Keyword(s):  
Simple Connectivity
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