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Geometric cycles and the cohomology of arithmetic subgroups of the exceptional group G2

We study the cohomology of a compact locally symmetric space attached to an arithmetic subgroup of a rational form of a group of type G2. By constructing suitable geometric cycles, we prove non‐vanishing results for this cohomology. These cycles represent cohomology classes that are non‐invariant under the ambient real Lie group. By this construction, we cover all possible non‐vanishing results compatible with the known theorems regarding unitary representations with non‐zero cohomology. A decisive tool in our approach is an extension of a formula for the intersection numbers of two geometric cycles.