Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 5 (2012) 77–97 Asymptotic enumeration of reversible maps regardless of genus Michael Drmota ∗ Institute of Discrete Mathematics and Geometry, Vienna University of Technology Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria Roman Nedela † Matej Bel University and Mathematical Institute, Slovak Academy of Sciences Bansk´ a Bystrica, Slovakia. Received 31 January 2010, accepted 18 April 2011, published online 12 December 2011 Abstract We derive asymptotic expansions for the numbers U (n) of isomorphism classes of sensed maps on orientable surfaces with given number of edges n, where we do not specify the genus and for the numbers A(n) of reflexible maps with n edges. As expected the ratio A(n)/U (n) → 0 for n →∞. This shows that almost all maps are chiral. Moreover, we show log A(n) ∼ 1 2 log U (n) ∼ (n/2) log n. Due to a correspondence between sensed maps with given number of edges and torsion-free subgroups of the group Γ= 〈x, y|y 2 = 1〉 of given index, the obtained results give an information on asymptotic expansions for the number of conjugacy classes of such subgroups of given index. Keywords: Graph, map, enumeration, asymptotic. Math. Subj. Class.: 05C30,05A16 1 Introduction A map is a 2-cell decomposition of a compact connected surface. In this paper we prefer- ably consider maps on orientable surfaces without boundary. A sensed (respectively un- sensed) map is an equivalence class of maps on a closed orientable surface, where two maps * This author is supported by the Austrian Science Foundation FWF, project S9604, part of the Austrian Na- tional Research Network “Analytic Combinatorics and Probabilistic Number Theory”. † This author is supported by the national grants APVV-0223-10, VEGA and the Centre of Excellence in Information Systems CAKS, grant no. ITMS NFP26220120023. E-mail addresses: michael.drmota@tuwien.ac.at (Michael Drmota), nedela@savbb.sk (Roman Nedela) Copyright c  2012 DMFA Slovenije