Insightful question! The answer is yes - that something is called the "impedance" of the material, which is analogous to electrical resistance (or "friction" if you prefer): informally, how "hard" it is for a wave to travel in that medium. To fully characterize how the differences in impedance at an interface between materials changes the path of a wave you would have to solve Maxwell's equations in 2D, but fortunately smarter people have done that for you, and the solution reduces to Snell's law: n1 sin theta1 = n2 sintheta2 => sin theta1 = (n2/n1) sin theta1. The ratio of the refractive indices n2/n1 is equal to the ratio of the impedances Z1/Z2 (note they are flipped). Let us assume that n1=1, the vacuum, so Z1 is the impedance of free space. This means the smaller Z2 is, the smaller theta2 is. We can look up these impedances, and it turns out Z1 ~ 377 ohms, Z2 ~ 200 ohms. So θ2 is smaller than theta1, meaning light must bend towards the normal.
What about the shortening of wavelengths? Well again we have n1 c = n2 v so v = (n1/n2) c => lambda2 f2 = (n1/n2) lambda1 f1. The fs are equal so we cancel them and we get lambda2 = (n1/n2) lambda1 = (Z2/Z1) lambda1. So since Z2 is smaller than Z1 we also predict a shorter wavelength lambda2 than lambda1
