The Farrell-Tate and Bredon homology for PSL_4(Z) and other arithmetic groups
Résumé
We provide some new computations of Farrell–Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell-Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise. We provide an algorithm computing an efficient subdivision of a complex to achieve this rigidity property. Applying this algorithm to available cell complexes for PSL_4(Z) and PGL_3(Z[i]) provides computations of Farrell-Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring. On the other hand, in order to check correctness of the computer calculations, we describe the Farrell-Tate cohomology in some rank-one cases, using Brown's complex and a number-theoretic description of the conjugacy classification of cyclic subgroups.
Domaines
K-théorie et homologie [math.KT]
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FarellAndBredonHomologyOfPSL4ZAndOtherArithmeticGroups.pdf (326.61 Ko)
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