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1. CMB 2001 (vol 44 pp. 385)

Ballantine, Cristina M.
A Hypergraph with Commuting Partial Laplacians
Let $F$ be a totally real number field and let $\GL_{n}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

Keywords:Hecke operators, buildings
Categories:11F25, 20F32

2. CMB 1997 (vol 40 pp. 330)

Kapovich, Ilya
Amalgamated products and the Howson property
We show that if $A$ is a torsion-free word hyperbolic group which belongs to class $(Q)$, that is all finitely generated subgroups of $A$ are quasiconvex in $A$, then any maximal cyclic subgroup $U$ of $A$ is a Burns subgroup of $A$. This, in particular, implies that if $B$ is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then $A\ast_U B$, $\langle A,t \mid U^t=V\rangle$ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting $3$-manifold groups are known to belong to class $(Q)$ and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.

Categories:20E06, 20E07, 20F32

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