Three algebraic surfaces — the slowness surface, the phase-velocity surface, and the group-velocity surface — play fundamental roles in the theory of seismic wave propagation in anisotropic elastic media. While the slowness (sometimes called phase-slowness) and phase-velocity surfaces are fairly simple and their main algebraic properties are well understood, the group-velocity surfaces are extremely complex; they are complex to the extent that even the algebraic degree, D, of a system of polynomials describing the general group-velocity surface is currently unknown, and only the upper bound of the degree (D150) is available. This paper establishes the exact degree (D=86) of the general group-velocity surface along with two closely related to D quantities: the maximum number, B, of body waves that may propagate along a ray direction in a homogeneous anisotropic elastic solid (B=19) and the maximum number, H, of isolated, singularity-unrelated cusps of a group-velocity surface (H=16).

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