# 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 $n \geq 2$.

## 6 Citations

### A GENERALISATION OF HIGHER-RANK GRAPHS

- MathematicsBulletin of the Australian Mathematical Society
- 2021

Abstract We introduce ‘generalised higher-rank k-graphs’ as a class of categories equipped with a notion of size. They extend not only higher-rank k-graphs, but also the Levi categories introduced by…

### HIGHER RANK GRAPHS FROM CUBE COMPLEXES AND THEIR SPECTRAL THEORY

- Mathematics
- 2021

We propose constructions of k-graphs from combinatorial input and initiate a study of their spectral theory. Guided by geometric insight, we obtain several new series of k-graphs using cube complexes…

### C*-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory.

- Mathematics
- 2020

We unite elements of category theory, K-theory, and geometric group theory, by defining a class of groups called $k$-cube groups, which act freely and transitively on the product of $k$ trees, for…

### Higher dimensional digraphs from cube complexes and their spectral theory

- Mathematics
- 2021

. We deﬁne k -dimensional digraphs and initiate a study of their spectral theory. The k -dimensional digraphs can be viewed as generating graphs for small categories called k -graphs. Guided by…

### The Polycyclic Inverse Monoids and the Thompson Groups Revisited

- MathematicsSemigroups, Categories, and Partial Algebras
- 2021

We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of…

### Higman–Thompson‐like groups of higher rank graph C*‐algebras

- MathematicsBulletin of the London Mathematical Society
- 2022

Let Λ$\Lambda$ be a row‐finite and source‐free higher rank graph with finitely many vertices. In this paper, we define the Higman–Thompson‐like group Λht$\operatorname{\Lambda _{ht}}$ of the graph…

## References

SHOWING 1-10 OF 27 REFERENCES

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

- Mathematics
- 2004

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

- Mathematics
- 2000

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

- Mathematics
- 2011

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

- MathematicsInt. J. Algebra Comput.
- 2019

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

- MathematicsCanadian 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

- MathematicsJournal 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

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2013

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

- Mathematics
- 2000

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

- Mathematics
- 2007

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

- Mathematics
- 1999

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…