  Tens of thousands of red giant stars in the Kepler data exhibit solar-like
oscillations. Their oscillations enable us to study the internal physics from
core to surface, such as differential rotation. However, envelope rotation
rates have been measured for only a dozen RGB stars so far. The limited sample
hinders the theoretical interpretation of angular momentum transport in
post-main-sequence phases. We apply a new approach to calculate the asymptotic
frequencies of mixed modes, which accounts for the so-called near-degeneracy
effects and leads to more proper measurements of envelope rotation rates. By
fitting these asymptotic expressions to the observations, we obtain
measurements of the properties of g modes and mean core and envelope rotation
rates. Among 2495 stars with clear mixed-mode patterns, we found that 800 show
doublets and 1206 show triplets, doubling the size of pre-existing catalogues.
This led us to discover an over-density of stars that narrowly distribute
around a well-defined ridge in the plane showing core rotation rate versus
evolution along the RGB. With this work, we also increase the sample of stars
with measured envelope rotation rates by two orders of magnitude. We find a
decreasing trend between envelope rotation rates and evolution, implying that
the envelopes slow down with expansion, as expected. We find 243 stars whose
envelope rotation rates are significantly larger than zero. For these stars,
the core-to-envelope rotation ratios are around 20 and show a large spread with
evolution. Several stars show extremely mild differential rotations, with
core-to-surface ratios between 1 and 2. These stars also have very slow core
rotation rates, suggesting that they go through a peculiar rotational
evolution. We also discovered more stars located below the degeneracy sequence,
which will provide the opportunity to study the history of possible stellar
mergers.
