Assembly maps for topological cyclic homology of group algebras

We use assembly maps to study \mathbf{TC}(\mathbb{A}[G];p), the topological cyclic homology at a prime p of the group algebra of a discrete group G with coefficients in a connective ring spectrum \mathbb{A}. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map for the family of virtually cyclic subgroups is injective but in general not surjective.

Can Gerrymanders Be Detected? An Examination of Wisconsin’s State Assembly

In October 2017, the Supreme Court heard an appeal of a November 2016 ruling striking down Wisconsin’s State Assembly districts as a Republican gerrymander that illegally dilutes the weight of Democratic votes. We take the opportunity to revisit this litigation to evaluate three proposed methods of detecting gerrymanders: the “efficiency gap,” a count of Assembly districts carried by statewide candidates, and the difference between the district-level partisan median and mean. The first two measures figure either centrally or peripherally in the plaintiffs’ case in Wisconsin, while the third is the approach we favor. We expand on the analysis offered at trial by evaluating how these measures fare across a variety of elections in Wisconsin and with the aid of 10,000 alternative Assembly maps drawn by computer. The alternative maps provide the appropriate baseline with which to gauge the level of vote dilution in Wisconsin and distinguish between the effect of residential geography and the Legislature’s actions. The results show that Wisconsin’s Assembly map is a substantial gerrymander according the median–mean comparison across all elections, while the two tests relied upon by the plaintiffs provide mixed results. We examine the measurement qualities of each test and show that the efficiency gap and districts-carried count both capture elements beyond partisan bias. We find no similar ambiguity with the median–mean comparison and conclude that the plaintiffs’ claim that Wisconsin’s Assembly map systematically dilutes the weight of Democratic votes is correct.