Metric thickenings and group actions
Date
2020
Authors
Heim, Mark T., author
Adams, Henry, advisor
Peterson, Chris, advisor
Neilson, James, committee member
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Abstract
Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes.
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Subject
group action
projective space
Cech complex
Vietoris-Rips complex
metric thickening