55 research outputs found
Length spectra and degeneration of flat metrics
In this paper we consider flat metrics (semi-translation structures) on
surfaces of finite type. There are two main results. The first is a complete
description of when a set of simple closed curves is spectrally rigid, that is,
when the length vector determines a metric among the class of flat metrics.
Secondly, we give an embedding into the space of geodesic currents and use this
to get a boundary for the space of flat metrics. The geometric interpretation
is that flat metrics degenerate to "mixed structures" on the surface: part flat
metric and part measured foliation.Comment: 36 page
Rational growth in the Heisenberg group
A group presentation is said to have rational growth if the generating series
associated to its growth function represents a rational function. A
long-standing open question asks whether the Heisenberg group has rational
growth for all finite generating sets, and we settle this question
affirmatively. We also establish almost-convexity for all finite generating
sets. Previously, both of these properties were known to hold for hyperbolic
groups and virtually abelian groups, and there were no further examples in
either case. Our main method is a close description of the relationship between
word metrics and associated Carnot-Caratheodory Finsler metrics on the ambient
Lie group. We provide (non-regular) languages in any word metric that suffice
to represent all group elements.Comment: Version 2 contains bug fixes and an added application to
almost-convexit
Redistricting for Proportionality
American democracy is currently heavily reliant on plurality in single-member districts, or PSMD, as a system of election. But public perceptions of fairness are often keyed to partisan proportionality, or the degree of congruence between each party’s share of the vote and its share of representation. PSMD has not tended to secure proportional outcomes historically, partially due to gerrymandering, where line-drawers intentionally extract more advantage for their side. But it is now increasingly clear that even blind PSMD is frequently disproportional, and in unpredictable ways that depend on local political geography. In this paper we consider whether it is feasible to bring PSMD into alignment with a proportionality norm by targeting proportional outcomes in the design and selection of districts. We do this mainly through a close examination of the “Freedom to Vote Test,” a redistricting reform proposed in draft legislation in 2021. We find that applying the test with a proportionality target makes for sound policy: it performs well in legal battleground states and has a workable exception to handle edge cases where proportionality is out of reach
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