On the asymptotics of visible elements and homogeneous
equations in surface groups
| dc.contributor.author | Antolín Pichel, Yago | |
| dc.contributor.author | Ciobanu, Laura | |
| dc.contributor.author | Viles, Noelia | |
| dc.date.accessioned | 2023-06-20T01:23:43Z | |
| dc.date.available | 2023-06-20T01:23:43Z | |
| dc.date.issued | 2012 | |
| dc.description.abstract | Let F be a group whose abelianization is Zk, k 2. An element of F is called visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1. In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity. | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/76831 | |
| dc.identifier.doi | 10.4171/GGD/167 | |
| dc.identifier.issn | 1661-7207 | |
| dc.identifier.officialurl | https://doi.org/10.4171/GGD/167 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/43565 | |
| dc.journal.title | Groups, geometry and dynamics | |
| dc.language.iso | eng | |
| dc.page.final | 638 | |
| dc.page.initial | 619 | |
| dc.publisher | EMS Press | |
| dc.rights | Atribución 3.0 España | |
| dc.rights.accessRights | open access | |
| dc.rights.uri | https://creativecommons.org/licenses/by/3.0/es/ | |
| dc.subject.cdu | 512 | |
| dc.subject.keyword | Free groups | |
| dc.subject.keyword | Surface groups | |
| dc.subject.keyword | Equations | |
| dc.subject.keyword | Visible elements | |
| dc.subject.keyword | Asymptotic behavior. | |
| dc.subject.ucm | Álgebra | |
| dc.subject.ucm | Grupos (Matemáticas) | |
| dc.subject.unesco | 1201 Álgebra | |
| dc.title | On the asymptotics of visible elements and homogeneous equations in surface groups | |
| dc.type | journal article | |
| dc.volume.number | 6 | |
| dcterms.references | [1] G. N. Arzhantseva and A. Yu. Ol0 shanskii, The class of groups all of whose subgroups with lesser number of generators are free is generic. Mat. Zametki 59 (1996), 489–496; English transl. Math. Notes 59 (1996), 350–355. Zbl 0877.20021 MR 1445193 [2] P. Billingsley, Probability and measure. 2nd ed., John Wiley, New York 1986. Zbl 0649.60001 MR 830424 [3] D. Calegari and K. Fujiwara, Combable functions, quasimorphisms, and the central limit theorem. Ergodic Theory Dynam. Systems 30 (2010), 1343–1369. Zbl 1217.37025 MR 2718897 [4] J. Christopher, The asymptotic density of some k-dimensional sets. Amer. Math. Monthly 63 (1956), 399–401. Zbl 0070.04101 MR 0097363 [5] L. Ciobanu, Polynomial-time complexity for instances of the endomorphism problem in free groups. Internat. J. Algebra Comput. 17 (2007), 289–328. Zbl 1125.20018 MR 2310149 [6] N. M. Dunfield and D. P. Thurston, A random tunnel number one 3-manifold does not fiber over the circle. Geom. Topol. 10 (2006), 2431–2499. Zbl 1139.57018 MR 2284062 [7] R. H. Gilman, A. Myasnikov, and V. Roman’kov, Random equations in free groups. Groups Complex. Cryptol. 3 (2011), 257–284. Zbl 06036193 MR 2898893 [8] R. H. Gilman, A. Myasnikov, and V. Roman’kov, Random equations in nilpotent groups. J. Algebra 352 (2012), 192–214. Zbl 06055217 MR 2862182 [9] I. Kapovich, I. Rivin, P. Schupp, and V. Shpilrain, Densities in free groups and Zk, visible points and test elements. Math. Res. Lett. 14 (2007), 263–284. Zbl 1132.20017 MR 2318624 [10] I. Kapovich, P. Schupp, and V. Shpilrain, Generic properties of Whitehead’s algorithm and isomorphism rigidity of random one-relator groups. Pacific J. Math. 223 (2006), 113–140. Zbl 1149.20028 MR 2221020 [11] Y. N. Petridis and M. S. Risager, Equidistribution of geodesics on homology classes and analogues for free groups. Forum Math. 20 (2008), 783–815. Zbl 1166.53015 MR 2445118 [12] I. Rivin, Growth in free groups (and other stories)—twelve years later. Illinois J. Math. 54 (2010), 327–370. Zbl 1225.05128 MR 2776999 [13] R. Sharp, Local limit theorems for free groups. Math. Ann. 321 (2001), 889–904. Zbl 1012.20019 MR 1872533 [14] W. Woess, Cogrowth of groups and simple random walks. Arch. Math. (Basel) 41 (1983), 363–370. Zbl 0522.20043 MR 731608 [15] R. J. Zimmer, Ergodic theory and semisimple groups. Monogr. Math. 81, Birkhäuser, Basel 1984. Zbl 0571.58015 MR 776417 | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | bd3bab81-47d2-4551-811a-af8ac40597c5 | |
| relation.isAuthorOfPublication.latestForDiscovery | bd3bab81-47d2-4551-811a-af8ac40597c5 |
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