Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms

Duke Mathematical Journal

Volumn 142, Issue 2, 2008, Pages 353-379

  Rivin, Igor ^a  

^a Temple University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 42549120299     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/00127094-2008-009     Document Type: Article

References (37)

1. N. H. ABEL, Beweis der Unmöglichkeit, algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzulösen, J. Reine Angew. Math. 1 (1826), 65-84.
2. M. BESTVINA and M. HANDEL, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), 1 -51. MR 1147956
3. A. BOREL, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991. MR 1102012
4. M. R. BRIDSON and K. VOGTMANN, "Automorphism groups of free groups, surface groups and free abelian groups" in Problems on Mapping Class Groups and Related Topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc, Providence, 2006,301-316. MR 2264548
5. A. J. CASSON and S. A. BLEILER, Automorphisms of Surfaces After Nielsen and Thurston, London Math. Soc. Stud. Texts 9, Cambridge Univ. Press, Cambridge, 1988. MR 0964685
6. N. CHAVDAROV, The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy, Duke Math. J. 87 (1997), 151-180. MR 1440067
7. D. COX, J. LITTLE, and D. O'SHEA, Ideals, Varieties, and Algorithms, 2nd ed., Undergrad. Texts Math., Springer, New York, 1997. MR 1417938
8. L. E. DICKSON, Theory of linear groups in an arbitrary field, Trans. Amer. Math. Soc. 2, no. 4 (1901), 363-394. MR 1500573
9. W. DUKE, Z. RUDNICK, and P. SARNAK, Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), 143-179. MR 1230289
10. A. ESKIN and C. MCMULLEN, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), 181 -209. MR 1230290
11. K. FORD, The distribution of totients, Ramanujan J. 2 (1998), 67-151. MR 1642874
12. _, The number of solutions of φ(x) = m, Ann. of Math. (2) 150 (1999), 283-311. MR 1715326
13. P. X. GALLAGHER, "The large sieve and probabilistic Galois theory" in Analytic Number Theory (St. Louis, Mo., 1972), Amer. Math. Soc, Providence, 1973, 91-101. MR 0332694
14. M. GECK, An Introduction to Algebraic Geometry and Algebraic Groups, Oxf. Grad. Texts Math. 10, Oxford Univ. Press, Oxford, 2003. MR 2032320
15. I. GELFAND, Normierte Ringe, Rec. Math. [Mat. Sbornik] N. S. 9 (1941), 3-24. MR 0004726
16. M. HALL JR., The Theory of Groups, Macmillan, New York, 1959. MR 0103215
17. M. HANDEL and L. MOSHER, Parageometric outer automorphisms of free groups, Trans. Amer. Math. Soc. 359, no. 7 (2007), 3153-3183. MR 2299450
18. _, The expansion factors of an outer automorphism and its inverse, Trans. Amer. Math. Soc. 359, no. 7 (2007), 3185-3208. MR 2299451
19. R. A. HORN and C. R. JOHNSON, Matrix Analysis, corrected reprint of the 1985 original, Cambridge Univ. Press, Cambridge, 1990. MR 1084815
20. I. KAPOVICH, personal communication, Nov. 2005.
21. D. KIRBY, Integer matrices of finite order, Rend. Mat. (6) 2 (1969), 403-408. MR 0258855
22. E. KOWALSKI, The principle of the large sieve, preprint, arXiv:math/0610021v3 [math.NT]
23. S. LANG, Algebra, 2nd ed., Addison-Wesley, Reading, Mass., 1984. MR 0783636
24. S. LANG and A. WEIL, Number of points of varieties infinite fields, Amer. J. Math. 76 (1954), 819-827. MR 0065218
25. J. MAHER, Random walks on the mapping class group, preprint, arXiv:math/0604433v2 [math.GT]
26. H. MAIER and C. POMERANCE, On the number of distinct values of Euler's φ-function, Acta Arith. 49 (1988), 263-275. MR 0932525
27. L. MOSHER, Mapping class groups are automatic, Ann. of Math. (2) 142 (1995), 303-384. MR 1343324
28. _, personal communication, 2007.
29. A. NEVO and P. SARNAK, Prime integral matrices, preprint, 2006.
30. M. NEWMAN, Integral Matrices, Pure Appl. Math. 45, Academic Press, New York, 1972. MR 0340283
31. I. RIVIN, Growth in free groups (and other stories), preprint, arXiv:math/9911076v2 [math.CO]
32. _, Walks on graphs and lattices - Effective bounds and applications, preprint, arXiv:math/0703533vl [math.NT]
33. J.-P. SERRE, Représentations linéaires des groupes finis, 3rd. ed., Hermann, Paris, 1978. MR 0543841
34. _, On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 429 - 440. MR 1997347
35. B. SIMON, Representations of Finite and Compact Groups, Grad. Stud. Math. 10, Amer. Math. Soc, Providence, 1996. MR 1363490
36. A. WEIL, Sur les courbes algébriques et les variétés qui s' en déduisent, Actualités Sci. Ind. 1041, Publ. Inst. Math. Univ. Strasbourg 7, Hermann, Paris, 1948. MR 0027151
37. H. WIELANDT, Finite Permutation Groups, Academic Press, New York, 1964. MR 0183775