Geodesic
The extrinsic primitive torsion problem
Bou-Rabee, Khalid1 ; Hooper, W Patrick1
1Department of Mathematics, The City College of New York, New York, NY, United States, Department of Mathematics, The Graduate Center, CUNY, New York, NY, United States
Algebraic and Geometric Topology, Tome 20 (2020) no. 7, pp. 3329-3376
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Let Pk be the subgroup generated by k th powers of primitive elements in Fr, the free group of rank r. We show that F2∕Pk is finite if and only if k is 1, 2 or 3. We also fully characterize F2∕Pk for k = 2,3,4. In particular, we give a faithful 9–dimensional representation of F2∕P4 with infinite image.

DOI : 10.2140/agt.2020.20.3329
Classification : 20F05, 20F65, 20F38
Keywords: Burnside problem, primitive elements, characteristic subgroups, square-tiled surface

Bou-Rabee, Khalid  1   ; Hooper, W Patrick  1

1 Department of Mathematics, The City College of New York, New York, NY, United States, Department of Mathematics, The Graduate Center, CUNY, New York, NY, United States 
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Bou-Rabee, Khalid; Hooper, W Patrick. The extrinsic primitive torsion problem. Algebraic and Geometric Topology, Tome 20 (2020) no. 7, pp. 3329-3376. doi: 10.2140/agt.2020.20.3329

[1] H Bradford, A Thom, Short laws for finite groups and residual finiteness growth, Trans. Amer. Math. Soc. 371 (2019) 6447 | DOI

[2] W Burnside, On an unsettled question in the theory of discontinuous groups, Q. J. Pure Appl. Math. 33 (1902) 230

[3] A E Clement, S Majewicz, M Zyman, The theory of nilpotent groups, Birkhäuser (2017) | DOI

[4] J P Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation, from: "Ergodic theory" (editor I Assani), Contemp. Math. 485, Amer. Math. Soc. (2009) 45 | DOI

[5] R Coulon, D Gruber, Small cancellation theory over Burnside groups, Adv. Math. 353 (2019) 722 | DOI

[6] G Forni, On the Lyapunov exponents of the Kontsevich–Zorich cocycle, from: "Handbook of dynamical systems, I-B" (editors B Hasselblatt, A Katok), Elsevier (2006) 549 | DOI

[7] G Forni, C Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich–Zorich spectrum, preprint (2008)

[8] G Forni, C Matheus, A Zorich, Square-tiled cyclic covers, J. Mod. Dyn. 5 (2011) 285 | DOI

[9] E S Golod, I R Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 261

[10] F Herrlich, Teichmüller curves defined by characteristic origamis, from: "The geometry of Riemann surfaces and abelian varieties" (editors J M Muñoz Porras, S Popescu, R E Rodríguez), Contemp. Math. 397, Amer. Math. Soc. (2006) 133 | DOI

[11] F Herrlich, G Schmithüsen, An extraordinary origami curve, Math. Nachr. 281 (2008) 219 | DOI

[12] W P Hooper, The invariant measures of some infinite interval exchange maps, Geom. Topol. 19 (2015) 1895 | DOI

[13] T Koberda, R Santharoubane, Quotients of surface groups and homology of finite covers via quantum representations, Invent. Math. 206 (2016) 269 | DOI

[14] G Kozma, A Thom, Divisibility and laws in finite simple groups, Math. Ann. 364 (2016) 79 | DOI

[15] J Malestein, A Putman, Simple closed curves, finite covers of surfaces, and power subgroups of Out(Fn), Duke Math. J. 168 (2019) 2701 | DOI

[16] C Matheus, G Weitze-Schmithüsen, Some examples of isotropic SL(2, R)–invariant subbundles of the Hodge bundle, Int. Math. Res. Not. 2015 (2015) 8657 | DOI

[17] A Y Olshansky, An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) 309

[18] T G Group, GAP: groups, algorithms, and programming, software (2017)

[19] T S Developers, SageMath, software (2019)

[20] A Zorich, Flat surfaces, from: "Frontiers in number theory, physics, and geometry, I" (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer (2006) 437

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