{"paper":{"title":"Non-injective representations of a closed surface group into $PSL(2,\\mathbb R)$","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Louis Funar, Maxime Wolff","submitted_at":"2005-02-28T17:45:44Z","abstract_excerpt":"Let $e$ denote the Euler class on the space $Hom(\\Gamma_g, PSL(2,\\mathbb R))$ of representations of the fundamental group $\\Gamma_g$ of the closed surface $\\Sigma_g$ of genus $g$. Goldman showed that the connected components of $Hom(\\Gamma_g, PSL(2,\\mathbb R))$ are precisely the inverse images $e^{-1}(k)$, for $2-2g\\leq k\\leq 2g-2$, and that the components of Euler class $2-2g$ and $2g-2$ consist of the injective representations whose image is a discrete subgroup of $PSL(2,\\mathbb R)$. We prove that non-faithful representations are dense in all the other components. We show that the image of a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502585","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}