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We study the homology and the Borel-Moore homology with coefficients in$\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Titsbuilding of $\mathrm{PGL}$ of a nonarchimedean local field of positivecharacteristic by an arithmetic subgroup (a special case of the generaldefinition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)).  We define an analogue of modular symbols in this context and show that theimage of the canonical map from homology to Borel-Moore homology is containedin the sub $\mathbb{Q}$-vector space generated by the modular symbols.  By definition, the limit of the Borel-Moore homology as the arithmetic groupbecomes small is isomorphic to the space of $\mathbb{Q}$-valued automorphicforms that satisfy certain conditions at a distinguished (fixed) place (namelythat it is fixed by the Iwahori subgroup and the center at the place). We showthat the limit of the homology with $\mathbb{C}$-coefficients is identifiedwith the subspace consisting of cusp forms. We also describe an irreduciblesubquotient of the limit of Borel-Moore homology as an induced representationin a precise manner and give a multiplicity one type result.

The Borel-Moore homology of an arithmetic quotient of the Bruhat-Tits  building of PGL of a non-archimedean local field in positive characteristic  and modular symbols