arXiv subjects
Publications and source records attributed to Guyan Robertson.
Let $Γ$ be an $\widetilde A_n$ subgroup of $PGL_{n+1}(K)$, with $n\ge 2$, where $K$ is a local field with residue field of order $q$ and let $\bb P^n_{K}$ be projective $n$-space over $K$. The module of coinvariants $H_0(Γ; C(P^n_{K},Z))$ is shown to be finite. Consequently there is no nonzero $Γ$-invariant $Z$-valued distribution on $P^n_{K}$.
Let $G$ be a connected semisimple real algebraic group. Assume that $G(\bb R)$ has no compact factors and let $Γ$ be a torsion-free uniform lattice subgroup of $G(\bb R)$. Then $Γ$ contains a malnormal abelian subgroup $A$. This implies that the $\tto$ factor $\vn(Γ)$ contains a masa $\fk A$ with Pukánszky invariant $\{\infty\}$.
Let $Γ$ be the fundamental group of a finite connected graph $\mathcal G$. Let $\mathfrak M$ be an abelian group. A {\it distribution} on the boundary $\partialΔ$ of the universal covering tree $Δ$ is an $\mathfrak M$-valued measure defined on clopen sets. If $\mathfrak M$ has no $χ(\mathcal G)$-torsion then the group of $Γ$-invariant distributions on $\partialΔ$ is isomorphic to $H_1(\mathcal G,\mathfrak M)$.
Let $Γ$ be a torsion free cocompact lattice in $\aut(\cl T_1)\times\aut(\cl T_2)$, where $\cl T_1$, $\cl T_2$ are trees whose vertices all have degree at least three. The group $H_2(Γ, \bb Z)$ is determined explicitly in terms of an associated 2-dimensional tiling system. It follows that under appropriate conditions the crossed product $C^*$-algebra $\cl A$ associated with the action of $Γ$ on the boundary of $\cl T_1\times \cl T_2$ satisfies $\rank K_0(\cl A) = 2\cdot\rank H_2(Γ, \bb Z)$.
Consider a compact locally symmetric space $M$ of rank $r$, with fundamental group $Γ$. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions $f\in\ell_2(Γ)$ which act by left convolution on $\ell_2(Γ)$. Let $T^r$ be a totally geodesic flat torus of dimension $r$ in $M$ and let $Γ_0\cong\bb Z^r$ be the image of the fundamental group of $T^r$ in $Γ$. Then $\vn(Γ_0)$ is a maximal abelian $\star$-subalgebra of $\vn(Γ)$ and its unitary normalizer is as small as possible. If $M$ has constant negative curvature then the Pukánszky invariant of $\vn(Γ_0)$ is $\infty$.
Let $(G,{\mathfrak I},N,S)$ be an affine topological Tits system, and let $Γ$ be a torsion free cocompact lattice in $G$. This article studies the coinvariants $H_0(Γ; C(Ω,{\mathbb Z}))$, where $Ω$ is the Furstenberg boundary of $G$. It is shown that the class $[1]$ of the identity function in $H_0(Γ; C(Ω,{\mathbb Z}))$ has finite order, with explicit bounds for the order. A similar statement applies to the $K_0$ group of the boundary crossed product $C^*$-algebra $C(Ω)\rtimesΓ$. If the Tits system has type $\widetilde A_2$, exact computations are given, both for the crossed product algebra and for the reduced group $C^*$-algebra.
A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.
Let $X$ be a finite connected graph, each of whose vertices has degree at least three. The fundamental group $Γ$ of $X$ is a free group and acts on the universal covering tree $Δ$ and on its boundary $\partial Δ$, endowed with a natural topology and Borel measure. The crossed product $C^*$-algebra $C(\partial Δ) \rtimes Γ$ depends only on the rank of $Γ$ and is a Cuntz-Krieger algebra whose structure is explicitly determined. The crossed product von Neumann algebra does not possess this rigidity. If $X$ is homogeneous of degree $q+1$ then the von Neumann algebra $L^\infty(\partial Δ)\rtimes Γ$ is the hyperfinite factor of type $III_λ$ where $λ=1/{q^2}$ if $X$ is bipartite, and $λ=1/{q}$ otherwise.
Let $Γ$ be a torsion free lattice in $G=\PGL(3,{\mathbb F})$ where ${\mathbb F}$ is a nonarchimedean local field. Then $Γ$ acts freely on the affine Bruhat-Tits building ${\mathcal B}$ of $G$ and there is an induced action on the boundary $Ω$ of ${\mathcal B}$. The crossed product $C^*$-algebra ${\mathcal A}(Γ)=C(Ω) \rtimes Γ$ depends only on $Γ$ and is classified by its K-theory. This article shows how to compute the K-theory of ${\mathcal A}(Γ)$ and of the larger class of rank two Cuntz-Krieger algebras.
Let $Γ$ be a torsion free lattice in $G = \PGL(n+1,\FF)$, where $n\ge 1$ and $\FF$ is a non-archimedean local field. Then $Γ$ acts on the Furstenberg boundary $G/P$, where $P$ is a minimal parabolic subgroup of $G$. The identity element $\id$ in the crossed product $C^*$-algebra $C(G/P)\rtimes Γ$ generates a class $[\id]$ in the $K_0$ group of $C(G/P)\rtimes Γ$. It is shown that $[\id]$ is a torsion element of $K_0$ and there is an explicit bound for the order of $[\id]$. The result is proved more generally for groups acting on affine buildings of type $\tA_n$. For $n=1, 2$ the Euler-Poincaré characteristic $χ(Γ)$ annihilates the class $[\id]$.
This is an appendix to the paper {\bf Asymptotic K-theory for groups acting on $\tA_2$ buildings}, and contains the results of the computations performed by the authors.
Let $\G$ be a group of type rotating automorphisms of an affine building $\cB$ of type $\wt A_2$. If $\G$ acts freely on the vertices of $\cB$ with finitely many orbits, and if $Ω$ is the (maximal) boundary of $\cB$, then $C(\Om)\rtimes \G$ is a p.i.s.u.n. $C^*$-algebra. This algebra has a structure theory analogous to that of a simple Cuntz-Krieger algebra and is the motivation for a theory of higher rank Cuntz-Krieger algebras, which has been developed by T. Steger and G. Robertson. The K-theory of these algebras can be computed explicitly in the rank two case. For the rank two examples of the form $C(\Om)\rtimes \G$ which arise from boundary actions on $\wt A_2$ buildings, the two K-groups coincide.