Cancer sequencing predicts driver genes using recurrent protein-altering mutations, but detecting recurrence for non-coding mutations remains unsolved. Here, we present a convergence framework for recurrence analysis of non-coding mutations, using three-dimensional co-localization of epigenomically-defined regions. We define the regulatory plexus of each gene as its cell-type-specific three-dimensional gene-regulatory neighborhood, inferred using Hi-C chromosomal interactions and chromatin state annotations. Using 16 matched tumor-normal prostate transcriptomes, we predict tumor-upregulated genes, and find enriched plexus mutations in distal regulatory regions normally repressed in prostate, suggesting out-of-context de-repression. Using 55 matched tumor-normal prostate genomes, we predict 15 driver genes by convergence of dispersed, low-frequency mutations into high-frequency dysregulatory events along prostate-specific plexi, controlling for mutational heterogeneity across regions, chromatin states, and patients. These play roles in growth signaling, immune evasion, mitochondrial function, and vascularization, suggesting higher-order pathway-level convergence. We experimentally validate the PLCB4 plexus and its ability to affect the canonical PI3K cancer pathway.