On asymptotic dimension of amalgamated products and right-angled Coxeter groups

This chapter considers the notion of asymptotic dimension. It first provides an overview of asymptotic dimension before explaining the asymptotic dimensions of free abelian and nonabelian groups, free groups, groups, integers, metric spaces, and products. It then looks at the large-scale geometry of a Cayley graph of a finitely generated group, along with the topological notion of dimension and the large-scale notion of dimension. It also analyzes three typical questions that are asked about asymptotic dimension before concluding with some other examples of groups where something is known about their asymptotic dimension, such as braid groups, Artin groups, surface groups, hyperbolic groups, Coxeter groups, Thompson's group, and wreath product. The discussion includes exercises and research projects.

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Automorphisms of odd Coxeter groups

Monatshefte für Mathematik ◽

10.1007/s00605-020-01496-3 ◽

2021 ◽

Author(s):

Tushar Kanta Naik ◽

Mahender Singh

Keyword(s):

Coxeter Groups

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The 300 “correlators” suggests 4D, $$ \mathcal{N} $$ = 1 SUSY is a solution to a set of Sudoku puzzles

Journal of High Energy Physics ◽

10.1007/jhep05(2021)077 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Aleksander J. Cianciara ◽

S. James Gates ◽

Yangrui Hu ◽

Renée Kirk

Keyword(s):

Coxeter Groups ◽

Auxiliary Field ◽

Permutation Groups ◽

Weight Space ◽

Field Problem ◽

Truncated Octahedron ◽

Color Problem ◽

Minimal Representations

Abstract A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ N -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ N = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

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