### Kazhdan constants and matrix coefficients of $\text{Sp}(n,\mathbb{R})$.

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We discuss Bass's conjecture on the vanishing of the Hattori-Stallings rank from the viewpoint of geometric group theory. It is noted that groups without u-elements satisfy this conjecture. This leads in particular to a simple proof of the conjecture in the case of groups of subexponential growth.

Let ${T}_{1},\cdots ,{T}_{d}$ be homogeneous trees with degrees ${q}_{1}+1,\cdots ,{q}_{d}+1\ge 3$, respectively. For each tree, let $\U0001d525:{T}_{j}\to \mathbb{Z}$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of ${T}_{1},\cdots ,{T}_{d}$ is the graph $\mathrm{\U0001d5a3\U0001d5ab}({q}_{1},\cdots ,{q}_{d})$ consisting of all $d$-tuples ${x}_{1}\cdots {x}_{d}\in {T}_{1}\times \cdots \times {T}_{d}$ with $\U0001d525\left({x}_{1}\right)+\cdots +\U0001d525\left({x}_{d}\right)=0$, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If $d=2$ and ${q}_{1}={q}_{2}=q$ then we obtain a Cayley graph of the...

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