The geometry of the 2+1 black hole in Einstein gravity with a negative cosmological constant is analyzed. The black hole arises from identifications of anti-de Sitter space by a discrete subgroup of its symmetry group SO(2,2). The black hole is a smooth manifold with a curvature singularity at r=0, which is not a metric singularity but a causal structure singularity. The metric is regular at r=0 but becomes unstable when coupled to matter. Kruskal coordinates and Penrose diagrams are introduced to study the causal structure. The black hole is shown to be similar to the Kerr solution in 3+1 dimensions, with an ergosphere and an upper bound on angular momentum. The black hole can be obtained by identifying anti-de Sitter space with a discrete subgroup of its symmetry group. The black hole has two Killing vectors, and its global structure is similar to the 3+1 case. The Penrose diagrams show the maximal causal extension of the black hole, with no closed timelike curves. The black hole has a mass and angular momentum, and its geometry is described by the metric (2.7). The black hole is a solution of the source-free Einstein equations everywhere, including r=0. The surface r=0 is not a conical singularity but a singularity in the causal structure. The black hole has a horizon at r=r_+, an ergosphere between r_+ and r_erg, and a surface of infinite redshift at r_erg. The black hole can be obtained by identifying anti-de Sitter space with a discrete subgroup of its symmetry group. The black hole has two Killing vectors, and its global structure is similar to the 3+1 case. The Penrose diagrams show the maximal causal extension of the black hole, with no closed timelike curves. The black hole has a mass and angular momentum, and its geometry is described by the metric (2.7). The black hole is a solution of the source-free Einstein equations everywhere, including r=0. The surface r=0 is not a conical singularity but a singularity in the causal structure. The black hole has a horizon at r=r_+, an ergosphere between r_+ and r_erg, and a surface of infinite redshift at r_erg. The black hole can be obtained by identifying anti-de Sitter space with a discrete subgroup of its symmetry group. The black hole has two Killing vectors, and its global structure is similar to the 3+1 case. The Penrose diagrams show the maximal causal extension of the black hole, with no closed timelike curves. The black hole has a mass and angular momentum, and its geometry is described by the metric (2.7). The black hole is a solution of the source-free Einstein equations everywhere, including r=0. The surface r=0 is not a conical singularity but a singularity inThe geometry of the 2+1 black hole in Einstein gravity with a negative cosmological constant is analyzed. The black hole arises from identifications of anti-de Sitter space by a discrete subgroup of its symmetry group SO(2,2). The black hole is a smooth manifold with a curvature singularity at r=0, which is not a metric singularity but a causal structure singularity. The metric is regular at r=0 but becomes unstable when coupled to matter. Kruskal coordinates and Penrose diagrams are introduced to study the causal structure. The black hole is shown to be similar to the Kerr solution in 3+1 dimensions, with an ergosphere and an upper bound on angular momentum. The black hole can be obtained by identifying anti-de Sitter space with a discrete subgroup of its symmetry group. The black hole has two Killing vectors, and its global structure is similar to the 3+1 case. The Penrose diagrams show the maximal causal extension of the black hole, with no closed timelike curves. The black hole has a mass and angular momentum, and its geometry is described by the metric (2.7). The black hole is a solution of the source-free Einstein equations everywhere, including r=0. The surface r=0 is not a conical singularity but a singularity in the causal structure. The black hole has a horizon at r=r_+, an ergosphere between r_+ and r_erg, and a surface of infinite redshift at r_erg. The black hole can be obtained by identifying anti-de Sitter space with a discrete subgroup of its symmetry group. The black hole has two Killing vectors, and its global structure is similar to the 3+1 case. The Penrose diagrams show the maximal causal extension of the black hole, with no closed timelike curves. The black hole has a mass and angular momentum, and its geometry is described by the metric (2.7). The black hole is a solution of the source-free Einstein equations everywhere, including r=0. The surface r=0 is not a conical singularity but a singularity in the causal structure. The black hole has a horizon at r=r_+, an ergosphere between r_+ and r_erg, and a surface of infinite redshift at r_erg. The black hole can be obtained by identifying anti-de Sitter space with a discrete subgroup of its symmetry group. The black hole has two Killing vectors, and its global structure is similar to the 3+1 case. The Penrose diagrams show the maximal causal extension of the black hole, with no closed timelike curves. The black hole has a mass and angular momentum, and its geometry is described by the metric (2.7). The black hole is a solution of the source-free Einstein equations everywhere, including r=0. The surface r=0 is not a conical singularity but a singularity in