Pared hyperbolic structures and volume

A compact orientable three-manifold $M$ is irreducible if every embedded $2$-sphere in its interior bounds an embedded $3$-ball in $M$. A properly embedded surface is incompressible if it admits no embedded disk whose interior misses the surface and whose boundary is an essential simple closed curve in it. A boundary subsurface is ${\pi }_1$-injective if the inclusion of each component induces an injection on fundamental groups. A pared pair is a pair $(M,P)$ in which $M$ is compact, orientable and irreducible, and $P$ is a possibly empty compact subsurface of $\mathrm{\partial }M$ whose pairwise disjoint components are ${\pi }_1$-injective annuli or tori, every noncyclic abelian subgroup of ${\pi }_1(M)$ is conjugate into ${\pi }_1$ of a component of $P$ up to base-path conjugacy, and every map of pairs

$$(S^1\times I,S^1\times \mathrm{\partial }I)⟶(M,P)$$

that induces injections on fundamental groups is homotopic through maps of pairs into $P$, so its boundary remains in $P$ during the homotopy. Write ${\mathrm{\partial }}_{0}M=\mathrm{\partial }M\setminus \mathrm{int}P$. A horoball is a region bounded by a horosphere, a Euclidean plane or sphere tangent to the ideal boundary at one point. The peripheral subgroup for a component of $P$ is the image of its fundamental group in ${\pi }_1(M)$, recorded up to conjugacy. A hyperbolic metric on $\mathrm{int}M$ is a curvature $-1$ metric described by an atlas of charts into ${\mathbb{H}}^3$ whose transition maps are in ${\mathrm{Isom}}^{+}({\mathbb{H}}^3)$. Its developing map $\mathrm{dev}$ from the universal cover and holonomy homomorphism $\rho :{\pi }_1(M)\to {\mathrm{Isom}}^{+}({\mathbb{H}}^3)$ satisfy $\mathrm{dev}(gx)=\rho (g)\mathrm{dev}(x)$. A hyperbolic structure on the pared pair $(M,P)$ is such a metric compatible with the pared marking: on the peripheral subgroup of each component of $P$, $\rho$ is injective and its image is conjugate to a discrete group of Euclidean translations fixing $\mathrm{\infty }$, of rank one for an annulus and rank two for a torus. These images preserve horoballs based at their fixed ideal point. This compatibility does not assert that an arbitrary incomplete hyperbolic metric has cusp neighborhoods. The metric is complete when it is geodesically complete. A torus cusp corresponding to a torus component of $P$ is an embedded end neighborhood isometric to a horoball quotient by that component's holonomy image, with its cross-section parallel to the marked boundary torus. A rank-one horoball quotient is the full cusp model for an annular peripheral subgroup; when a structure has totally geodesic boundary, an annular cusp uses the portion between the incident boundary planes and has annular cross-sections. The hyperbolic volume is the integral of the Riemannian volume density over $\mathrm{int}M$; the structure has finite volume when this integral is finite.

Ken’ichi Yoshida, The minimal volume orientable hyperbolic 3-manifold with 4 cusps, §2, Definition 2.1

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4