arXiv subjects
Publications and source records attributed to Guyan Robertson.
Let $Δ$ be a building of type $\widetilde A_2$ and order $q$, with maximal boundary $Ω$. Let $Γ$ be a group of type preserving automorphisms of $Δ$ which acts regularly on the chambers of $Δ$. Then the crossed product $C^*$-algebra $C(Ω) \rtimes Γ$ is isomorphic to $M_{3(q+1)} \otimes\cl O_{q^2}\otimes\cl O_{q^2}$, where $\cl O_n$ denotes the Cuntz algebra generated by $n$ isometries whose range projections sum to the identity operator.
Let $Δ$ be an infinite, locally finite tree with more than two ends. Let $Γ<\aut(Δ)$ be an acylindrical uniform lattice. Then the boundary algebra $\cl A_Γ= C(\partialΔ)\rtimes Γ$ is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.
We study certain group actions on triangle buildings and their boundaries and some von Neumann algebras which can be constructed from them. In particular, for buildings of order $q\geq 3$ certain natural actions on the boundary are hyperfinite of type $\tqs$.
A subgroup of an amenable group is amenable. The $C^*$-algebra version of this fact is false. This was first proved by M.-D. Choi who proved that the non-nuclear $C^*$-algebra $C^*_r(\ZZ_2*\ZZ_3)$ is a subalgebra of the nuclear Cuntz algebra ${\cal O}_2$. A. Connes provided another example, based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same. In fact he proved that certain of the $C^*$-algebras studied by J. Cuntz and W. Krieger [10] can be constructed naturally as crossed product algebras. For example if the group $Γ$ acts simply transitively on a homogeneous tree of finite degree with boundary $Ω$ then $\cross$ is a Cuntz-Krieger algebra. Such trees may be regarded as affine buildings of type $\widetilde A_1$. The present paper is devoted to the study of the analogous situation where a group $\G$ acts simply transitively on the vertices of an affine building of type $\widetilde A_2$ with boundary $Ø$. The corresponding crossed product algebra $\cross$ is then generated by two Cuntz-Krieger algebras. Moreover we show that $\cross$ is simple and nuclear. This is a consequence of the facts that the action of $\G$ on $Ø$ is minimal, topologically free, and amenable.
We construct factors of type $\tn$ for $n\in\NN, n\geq 2$ from group actions on homogeneous trees and their boundaries. Our result is a discrete analogue of a result of R.J Spatzier, where the hyperfinite factor of type $\tone$ is constructed from a group action on the boundary of the universal cover of a manifold.
An $\widetilde A_2$ group $Γ$ acts simply transitively on the vertices of an affine building $\triangle$. We study certain subgroups $Γ_0 \cong {\Bbb Z}^2$ which act on certain apartments of $\triangle$. If one of these subgroups acts simply transitively on an apartment, then the corresponding subalgebra of the group von Neumann algebra is maximal abelian and singular. Moreover the Pukánszky invariant contains a type $I_{\infty}$ summand.
Let $Γ$ be a group of type rotating automorphisms of a building $\cB$ of type $\widetilde A_2$, and suppose that $Γ$ acts freely and transitively on the vertex set of $\cB$. The apartments of $\cB$ are tiled by triangles, labelled according to $Γ$-orbits. Associated with these tilings there is a natural subshift of finite type, which is shown to be irreducible. The key element in the proof is a combinatorial result about finite projective planes.
In this paper we develop the theory of strongly singular subalgebras of von Neumann algebras, begun in earlier work. We mainly examine the situation of type $\tto$ factors arising from countable discrete groups. We give simple criteria for strong singularity, and use them to construct strongly singular subalgebras. We particularly focus on groups which act on geometric objects, where the underlying geometry leads to strong singularity.
Let $Γ$ be a group of type rotating automorphisms of a building $\fX$ of type $\tilde A_n$ and order $q$. Suppose that $\G$ acts freely and transitively on the vertex set of $\fX$. Then the action of $Γ$ on the boundary of $\fX$ is ergodic, of type $\tq$ or type $\tqs$ depending on whether $n$ is odd or even.
Let $T_1$ and $T_2$ be homogeneous trees of even degree $\ge 4$. A BM group $Γ$ is a torsion free discrete subgroup of $\aut (T_1) \times \aut (T_2)$ which acts freely and transitively on the vertex set of $T_1 \times T_2$. This article studies dynamical systems associated with BM groups. A higher rank Cuntz-Krieger algebra $\mathcal A(\G)$ is associated both with a 2-dimensional tiling system and with a boundary action of a BM group $Γ$. An explicit expression is given for the K-theory of $\mathcal A(\G)$. In particular $K_0=K_1$. A complete enumeration of possible BM groups $\G$ is given for a product homogeneous trees of degree 4, and the K-groups are computed.
A class of negative definite kernels is defined in terms of measure spaces. Using this concept, property (T) for a countable group $\G$ is characterized in terms of measure preserving actions of $\G$, as follows. If a set $S$ is translated a finite amount by any fixed element of $\G$, then there is a uniform bound on how far $S$ is translated.
The geodesic distance between points in real hyperbolic space is a hypermetric, and hence is a kernel negative type. The proof given here uses an integral formula for geodesic distance, in terms of a measure on the space of hyperplanes. An analogous integral formula, involving the space of horospheres, is given for complex hyperbolic space.By contrast geodesic distance in a projective space is not of negative type.
It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes.
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 particular, it is simple, purely infinite and nuclear. We study an example: if $\G$ is a group acting freely on the vertices of an $\wt A_2$ building, with finitely many orbits, and if $Ω$ is the boundary of that building, then $C(\Om)\rtimes \G$ is the algebra associated to a certain two dimensional subshift.
Haagerup's inequality for convolvers on free groups may be interpreted as a result on $\tA_1$ buildings, i.e. trees. Here are proved analogous inequalities for discrete groups acting freely on the vertices of $\tA_1\times\tA_1$ and $\tA_2$ buildings. The results apply in particular to groups of type-rotating automorphisms acting simply transitively on the vertices of such buildings. These results provide the first examples of higher rank groups with property (RD).
If $Γ$ is a torsion free $\widetilde A_2$ group acting on an $\widetilde A_2$ building $Δ$, and $\fk A_Γ$ is the associated boundary $C^*$-algebra, it is proved that $K_0(\fk A_Γ)\otimes \bb R \cong \bb R^{2β_2}$, where $β_2=\dim_\bb R H^2(Γ, \bb R)$.
Let $Γ$ be an $\widetilde A_2$ subgroup of $\PGL_3(\mathbb K)$, where $\mathbb K$ is a local field with residue field of order $q$. The module of coinvariants $C(\mathbb P^2_{\mathbb K},\mathbb Z)_Γ$ is shown to be finite, where $\mathbb P^2_{\mathbb K}$ is the projective plane over $\mathbb K$. If the group $Γ$ is of Tits type and if $q \not\equiv 1 \pmod {3}$ then the exact value of the order of the class $[I]_{K_0}$ in the K-theory of the (full) crossed product $C^*$-algebra $C(Ω)\rtimesΓ$ is determined, where $Ω$ is the Furstenberg boundary of $\PGL_3(\mathbb K)$. For groups of Tits type, this verifies a conjecture of G. Robertson and T. Steger.
Let $Γ$ be a torsion free discrete group acting cocompactly on a two dimensional euclidean building $Δ$. The centralizer of an element of $Γ$ is either a Bieberbach group or is described by a finite graph of finite cyclic groups. Explicit examples are computed, with $Δ$ of type $\widetilde A_2$.