Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization

For a geometrically finite Kleinian group Γ, the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow. Moreover, it is strongly mixing. We obtain a higher rank analogue of this theorem. Let Γ be a relatively Anosov subgroup of a semisimple real algebraic group G. For every linear form ψ tangent to the growth indicator of Γ, there is a canonical one-dimensional flow space (Ω_ψ, m_ψ, φₜ) such that the non-wandering set Ω_Γ for the diagonal flow is a ψ-vector bundle over Ω_ψ and the Bowen-Margulis-Sullivan measure on Ω_Γ is the product m_ψ ⊗ Lebψ. We show that mψ is of finite measure and is the unique measure of maximal entropy for the flow \φₜ\. Moreover, (Ωψ, mψ, φₜ) is strongly mixing. The main ingredient is our construction of a reparameterization of the flow \φₜ\ on Ωψ by the geodesic flow on the Groves-Manning cusp space of Γ, which has an exponentially expanding property along unstable foliations.