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

Sam A. Mutter, Aura-Cristiana Radu, Alina Vdovina

Abstract

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 arbitrary $k$. The quotient of this action on the product of trees defines a $k$-dimensional cube complex, which induces a higher-rank graph. We make deductions about the K-theory of the corresponding $k$-rank graph C*-algebras, and give explicit examples of $k$-cube groups and their K-theory. We give explicit computations of K-theory for an infinite family of $k$-rank graphs for $k\geq 3$, which is not a direct consequence of the Künneth Theorem for tensor products.

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

Abstract

We unite elements of category theory, K-theory, and geometric group theory, by defining a class of groups called -cube groups, which act freely and transitively on the product of trees, for arbitrary . The quotient of this action on the product of trees defines a -dimensional cube complex, which induces a higher-rank graph. We make deductions about the K-theory of the corresponding -rank graph C*-algebras, and give explicit examples of -cube groups and their K-theory. We give explicit computations of K-theory for an infinite family of -rank graphs for , which is not a direct consequence of the Künneth Theorem for tensor products.

Paper Structure

This paper contains 6 sections, 21 theorems, 85 equations, 8 figures.

Table of Contents

  1. Introduction
  2. $k$-cube groups
  3. Higher-rank graphs
  4. Spectral sequences and K-theory
  5. Examples for $k=3$ and $k=4$
  6. Higher-rank graphs arising as double covers of cube complexes

Key Result

Proposition 2.1

Let $\mathcal{M}$ be a $k$-dimensional cube complex. The universal cover of $\mathcal{M}$ is a product of $k$ trees $\tilde{\mathcal{M}} = T_1 \times \cdots \times T_k$ if and only if the link at each vertex of $\mathcal{M}$ is a clique complex of a complete $k$-partite graph.

Figures (8)

  • Figure 1: Given a pointed square $S = [a,b,a',b'] \in \mathcal{S}_2$, we write $S_H$, $S_R$, $S_V$ to denote the three pointed squares in $\mathcal{S}_2$ depicted above (compare with (\ref{['eq:squares']})). Geometrically, these squares are all in the same orbit under reflection in the horizontal and/or vertical directions, meaning they represent the same element in $\mathcal{S}_2'$. Figure \ref{['fig:cube_symmetries']} extends this notion to cubes.
  • Figure 2: For a pointed, oriented $3$-cube $S = [A,B,C,A',B',C'] \in \mathcal{S}_3$, the seven corresponding cubes from (\ref{['eq:cube_symmetries']}) are defined by reflecting and rotating $S$ according to the arrows above. The transformations map the original basepoint to a new vertex (blue), but the new cubes are given the same basepoint and orientation as $S$ (black).
  • Figure 3: Let $k \geq 4$. Above is depicted a pointed $4$-cube in $\mathcal{S}_4$, for some $k$-cube group with adjacency structure $E_1,\ldots , E_k$. Let $u^L$ be elements of $E_1$, and $v_i^L,w_i^L \in E_i$. Fix three mutually-adjacent squares: elements of $F(1,j)$ labelled $(u,v_j,u^j,w_j)$, for $j \in \lbrace 2,3,4\rbrace$. Then each of the remaining $u^L,v_i^L,w_i^L$ is uniquely-determined, such that they label the edges of the $4$-cube above. We have condensed the notation for the sets $L$ for clarity.
  • Figure 4: Geometric realisation of the pointed, oriented cube in the cube complex corresponding to $\mathcal{S}_3(\Gamma_{\lbrace 3,5,7\rbrace})$ labelled $[[a_1,b_1^{-1},a_2,b_3], [a_2,c_1^{-1},a_1^{-1},c_2], [b_2,c_2,b_1,c_1], \\ [a_2^{-1},b_3^{-1},a_2^{-1},b_2], [a_2,c_3,a_2,c_2^{-1}], [b_3,c_2^{-1},b_3^{-1},c_3^{-1}]].$The choice of basepoints and orientations of the cubes and their faces ($2$-cells) is arbitrary, but must remain consistent across the entire complex.
  • Figure 5: Let $a^L,u^L,x^L \in E_1$, $(\ast)_2^L \in E_2$, and $(\ast)_3^L \in E_3$, where $E_1,E_2,E_3$ form the adjacency structure of some $3$-cube group. Consider the pointed $3$-cubes $A,B,C \in \mathcal{S}_3$. If $B$ is $E_3$-adjacent to $A$ (resp. $C$ is $E_2$-adjacent to $A$), then the magenta (resp. blue) $2$-faces above coincide. In general, for a $k$-cube group, $M_i(A,B) = 1$ implies that some $(k-1)$-faces of the geometric realisations of $A$ and $B$ in the corresponding cube complex coincide.
  • ...and 3 more figures

Theorems & Definitions (66)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • ...and 56 more