• Corpus ID: 224709485

Higher dimensional generalizations of the Thompson groups via higher rank graphs

@article{Lawson2020HigherDG,
  title={Higher dimensional generalizations of the Thompson groups via higher rank graphs},
  author={Mark V. Lawson and Aidan Sims and Alina Vdovina},
  journal={arXiv: Group Theory},
  year={2020}
}
We construct a family of groups from suitable higher rank graphs which are analogues of the finite symmetric groups. We introduce homological invariants showing that many of our groups are, for example, not isomorphic to $nV$, when $n \geq 2$. 

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References

SHOWING 1-10 OF 27 REFERENCES

Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach

AbstractWe provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the

Higher Rank Graph C-Algebras

Building on recent work of Robertson and Steger, we associate a C{algebra to a combinatorial object which may be thought of as a higher rank graph. This C{algebra is shown to be isomorphic to that of

Groupoids and C * -algebras for categories of paths

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the constructions of algebras from directed graphs, higher-rank graphs, and

Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes

It is shown that vertex transitive lattices on products of trees of arbitrary dimension d ≥ 1 based on quaternion algebras over global fields with exactly two ramified places are constructed.

Gauge-Invariant Ideals in the C*-Algebras of Finitely Aligned Higher-Rank Graphs

  • A. Sims
  • Mathematics
    Canadian Journal of Mathematics
  • 2006
Abstract We produce a complete description of the lattice of gauge-invariant ideals in ${{C}^{*}}(\Lambda )$ for a finitely aligned $k$ -graph $\Lambda $ . We provide a condition on $\Lambda $ under

A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

  • M. Lawson
  • Mathematics
    Journal of the Australian Mathematical Society
  • 2010
Abstract We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and

Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Abstract Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the

On higher rank graph C ∗ -algebras

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C

Orthogonal Completions of the Polycyclic Monoids

We introduce the notion of an orthogonal completion of an inverse monoid with zero. We show that the orthogonal completion of the polycyclic monoid on n generators is isomorphic to the inverse monoid

Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras

To an $r$-dimensional subshift of finite type satisfying certain special properties we associate a $C^*$-algebra $\cA$. This algebra is a higher rank version of a Cuntz-Krieger algebra. In