Filter passband amplitude ripple and group delay ripple (group delay variation)
Whenever you have any complex function that is analytic in the upper half-plane (why this matters will become clear) you can relate uniquely and exactly the complex and real parts of the function through what is called the Kramers-Kronig relations. 

Why does this matter? Well, it turns out that any causal system, which is a system where the output depends only on past inputs (which any real-world system is unless you invent time-travel), is described with a complex function that is analytic in the upper half-plane. 

This means that if we only know the exact amplitude behavior of your system (which is the real part), we can calculate what the imaginary (phase/group delay) part must be. In other words, yes, there is a strict relationship between amplitude and group delay ripple of a filter.

This is also why it is impossible too make a filter (or any system really) that implements an arbitrary phase and amplitude response - if we pick one, we have indirectly also picked the other. 

This is sometimes used to calculate the loss of very, very low-loss materials, because it is much easier (but still really hard) to measure their phase behavior (which would be the permittivity/permeability) than it is to measure their loss. We can then calculate the loss from the phase behavior.