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Quotients of Buildings by Non-uniform Lattices
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abstract
We consider quotients of the Bruhat-Tits building associated to the projective linear groups of dimension $d>2$ over the function field $\mathbb F_q(t)$ by a non-uniform lattice $\Gamma$ which is a congruence subgroup in the non-uniform lattice $ PGL_{d}(R)$, where $R=\mathbb F_q[\frac{1}{t}]$. We determine a fundamental domain and demonstrate that the quotient, while not cofinite, is at least of finite covolume. We do the case $d=3$ in considerable detail.
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Cited by 1 Pith paper
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Vertex volumes, lattice-minima tails, and height zeta functions for the standard arithmetic quotient of $\operatorname{PGL}_d$
Dominant-sector coordinates give a closed vertex volume for Γ\PGL_d, sharp α∈L^r iff r<d cusp tails of order T^{-d}, and a rational height zeta with simple pole at s=d.
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