Abstract
We introduce a conjugation invariant normalized height $\widehat{h}(F)$ on finite subsets of matrices $F$ in $\mathrm{GL}_{d}(\overline{\Bbb{Q}})$ and describe its properties. In particular, we prove an analogue of the Lehmer problem for this height by showing that $\widehat{h}(F)>\varepsilon $ whenever $F$ generates a nonvirtually solvable subgroup of $\mathrm{GL}_{d}(\overline{\Bbb{Q}}),$ where $\varepsilon =\varepsilon (d)>0$ is an absolute constant. This can be seen as a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. As an application we prove a uniform version of the classical Burnside-Schur theorem on torsion linear groups. In a companion paper we will apply these results to prove a strong uniform version of the Tits alternative.
-
[AmoD]
F. Amoroso and S. David, "Le problème de Lehmer en dimension supérieure," J. Reine Angew. Math., vol. 513, pp. 145-179, 1999.
@article {AmoD, MRKEY = {1713323},
AUTHOR = {Amoroso, Francesco and David, Sinnou},
TITLE = {Le problème de {L}ehmer en dimension supérieure},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {513},
YEAR = {1999},
PAGES = {145--179},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11J25 (11G50)},
MRNUMBER = {1713323},
MRREVIEWER = {Paul M. Voutier},
DOI = {10.1515/crll.1999.058},
ZBLNUMBER = {1011.11045},
} -
[AmDv]
F. Amoroso and R. Dvornicich, "A lower bound for the height in abelian extensions," J. Number Theory, vol. 80, iss. 2, pp. 260-272, 2000.
@article {AmDv, MRKEY = {1740514},
AUTHOR = {Amoroso, Francesco and Dvornicich, Roberto},
TITLE = {A lower bound for the height in abelian extensions},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {80},
YEAR = {2000},
NUMBER = {2},
PAGES = {260--272},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11R18 (11G50)},
MRNUMBER = {1740514},
MRREVIEWER = {Christopher Smyth},
DOI = {10.1006/jnth.1999.2451},
ZBLNUMBER = {0973.11092},
} -
[AM] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Reading, MA: Addison-Wesley Publishing Co., 1969.
@book {AM, MRKEY = {0242802},
AUTHOR = {Atiyah, M. F. and Macdonald, I. G.},
TITLE = {Introduction to Commutative Algebra},
PUBLISHER = {Addison-Wesley Publishing Co.},
ADDRESS={Reading, MA},
YEAR = {1969},
PAGES = {ix+128},
MRCLASS = {13.00},
MRNUMBER = {0242802},
MRREVIEWER = {J. A. Johnson},
ZBLNUMBER = {0175.03601},
} -
[BakRum]
M. H. Baker and R. Rumely, "Equidistribution of small points, rational dynamics, and potential theory," Ann. Inst. Fourier $($Grenoble$)$, vol. 56, iss. 3, pp. 625-688, 2006.
@article {BakRum, MRKEY = {2244226},
AUTHOR = {Baker, Matthew H. and Rumely, Robert},
TITLE = {Equidistribution of small points, rational dynamics, and potential theory},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {56},
YEAR = {2006},
NUMBER = {3},
PAGES = {625--688},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {11G50 (37A45 37F10)},
MRNUMBER = {2244226},
URL = {http://aif.cedram.org/item?id=AIF_2006__56_3_625_0},
ZBLNUMBER = {05176555},
} -
[BarCor]
L. Bartholdi and Y. de Cornulier, "Infinite groups with large balls of torsion elements and small entropy," Arch. Math. $($Basel$)$, vol. 87, iss. 2, pp. 104-112, 2006.
@article {BarCor, MRKEY = {2249644},
AUTHOR = {Bartholdi, Laurent and de Cornulier, Yves},
TITLE = {Infinite groups with large balls of torsion elements and small entropy},
JOURNAL = {Arch. Math. $($Basel$)$},
FJOURNAL = {Archiv der Mathematik},
VOLUME = {87},
YEAR = {2006},
NUMBER = {2},
PAGES = {104--112},
ISSN = {0003-889X},
CODEN = {ACVMAL},
MRCLASS = {20F50 (20F16)},
MRNUMBER = {2249644},
MRREVIEWER = {Javier Otal},
DOI = {10.1007/s00013-005-1684-4},
ZBLNUMBER = {1136.20035},
} -
[Bilu]
Y. Bilu, "Limit distribution of small points on algebraic tori," Duke Math. J., vol. 89, iss. 3, pp. 465-476, 1997.
@article {Bilu, MRKEY = {1470340},
AUTHOR = {Bilu, Yuri},
TITLE = {Limit distribution of small points on algebraic tori},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {89},
YEAR = {1997},
NUMBER = {3},
PAGES = {465--476},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11G35 (11G25 11J68 14G05 14G25)},
MRNUMBER = {1470340},
MRREVIEWER = {Dan Abramovich},
DOI = {10.1215/S0012-7094-97-08921-3},
ZBLNUMBER = {0918.11035},
} -
[Bom]
E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Cambridge: Cambridge Univ. Press, 2006, vol. 4.
@book {Bom, MRKEY = {2216774},
AUTHOR = {Bombieri, Enrico and Gubler, Walter},
TITLE = {Heights in {D}iophantine Geometry},
SERIES = {New Math. Monogr.},
VOLUME = {4},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2006},
PAGES = {xvi+652},
ISBN = {978-0-521-84615-8; 0-521-84615-3},
MRCLASS = {11G50 (11-02 11G10 11G30 11J68 14G40)},
MRNUMBER = {2216774},
MRREVIEWER = {Yuri Bilu},
DOI = {10.1017/CBO9780511542879},
ZBLNUMBER = {1115.11034},
} -
[Bor] A. Borel, Linear Algebraic Groups, New York: W. A. Benjamin, 1969.
@book {Bor, MRKEY = {0251042},
AUTHOR = {Borel, Armand},
TITLE = {Linear Algebraic Groups},
NOTE = {notes taken by Hyman Bass},
PUBLISHER = {W. A. Benjamin},
ADDRESS={New York},
YEAR = {1969},
PAGES = {xi+398},
MRCLASS = {14.50},
MRNUMBER = {0251042},
MRREVIEWER = {F. D. Veldkamp},
ZBLNUMBER = {0186.33201},
} -
[Bourb] N. Bourbaki, Groupes et Algèbres de Lie, Hermann.
@book{Bourb,
author={Bourbaki, N.},
TITLE={Groupes et Algèbres de Lie},
PUBLISHER={Hermann},
NOTE={chapitres 4---8}
} -
[BreSol] E. Breuillard, "On uniform exponential growth for solvable groups," Pure Appl. Math. Q., vol. 3, iss. 4, part 1, pp. 949-967, 2007.
@article {BreSol, MRKEY = {2402591},
AUTHOR = {Breuillard, Emmanuel},
TITLE = {On uniform exponential growth for solvable groups},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {3},
YEAR = {2007},
NUMBER = {4, part 1},
PAGES = {949--967},
ISSN = {1558-8599},
MRCLASS = {20F16 (20F05 20F69)},
MRNUMBER = {2402591},
MRREVIEWER = {Goulnara N. Arzhantseva},
ZBLNUMBER = {1147.20027},
} -
[uti]
E. Breuillard and T. Gelander, "Uniform independence in linear groups," Invent. Math., vol. 173, iss. 2, pp. 225-263, 2008.
@article {uti, MRKEY = {2415307},
AUTHOR = {Breuillard, Emmanuel and Gelander, T.},
TITLE = {Uniform independence in linear groups},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {173},
YEAR = {2008},
NUMBER = {2},
PAGES = {225--263},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {20G30 (20F16 20F65 20G25 22E40)},
MRNUMBER = {2415307},
MRREVIEWER = {Herbert Abels},
DOI = {10.1007/s00222-007-0101-y},
ZBLNUMBER = {1148.20029},
} -
[Bre] E. Breuillard, "Heights on ${ {SL}}_{2}$ and free subgroups," in Geometry, Rigidity and Group Actions, Zimmer’s Festschrift, Farb, B. and Fisher, D., Eds., Chicago, IL: Chicago Univ. Press, 2011.
@incollection{Bre,
author = {Breuillard, Emmanuel},
TITLE={Heights on ${\rm {SL}}_{2}$ and free subgroups},
BOOKTITLE={Geometry, Rigidity and Group Actions, Zimmer's Festschrift},
EDITOR={Farb, B. and Fisher, D.},
PUBLISHER={Chicago Univ. Press},
ADDRESS={Chicago, IL},
YEAR={2011},
} -
[B] E. Breuillard, A strong Tits alternative, 2008.
@misc{B,
author = {Breuillard, Emmanuel},
TITLE={A strong {T}its alternative},
NOTE={preprint},
YEAR={2008},
ARXIV={0804.1395},
} -
[BreEffec] E. Breuillard, Effective estimates for the spectral radius of a finite set of matrices.
@misc{BreEffec, SORTYEAR={2020},
AUTHOR = {Breuillard, Emmanuel},
TITLE={Effective estimates for the spectral radius of a finite set of matrices},
NOTE={preprint},
} -
[BH] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, New York: Springer-Verlag, 1999, vol. 319.
@book {BH, MRKEY = {1744486},
AUTHOR = {Bridson, Martin R. and Haefliger, Andr{é}},
TITLE = {Metric Spaces of Non-Positive Curvature},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {319},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1999},
PAGES = {xxii+643},
ISBN = {3-540-64324-9},
MRCLASS = {53C23 (20F65 53C70 57M07)},
MRNUMBER = {1744486},
MRREVIEWER = {Athanase Papadopoulos},
ZBLNUMBER = {0988.53001},
} -
[BT]
F. Bruhat and J. Tits, "Groupes réductifs sur un corps local," Inst. Hautes Études Sci. Publ. Math., iss. 41, pp. 5-251, 1972.
@article {BT, MRKEY = {0327923},
AUTHOR = {Bruhat, F. and Tits, J.},
TITLE = {Groupes réductifs sur un corps local},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {41},
YEAR = {1972},
PAGES = {5--251},
ISSN = {0073-8301},
MRCLASS = {20G25 (22E20)},
MRNUMBER = {0327923},
MRREVIEWER = {M. Krusemeyer},
URL = {http://www.numdam.org/item?id=PMIHES_1972__41__5_0},
ZBLNUMBER = {0254.14017},
} -
[Cap] P. E. Caprace.
@misc{Cap,
author={Caprace, P. E.},
NOTE={personal communication},
} -
[Cham]
A. Chambert-Loir, "Mesures et équidistribution sur les espaces de Berkovich," J. Reine Angew. Math., vol. 595, pp. 215-235, 2006.
@article {Cham, MRKEY = {2244803},
AUTHOR = {Chambert-Loir, Antoine},
TITLE = {Mesures et équidistribution sur les espaces de {B}erkovich},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
VOLUME = {595},
YEAR = {2006},
PAGES = {215--235},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {14G40 (14G25)},
MRNUMBER = {2244803},
DOI = {10.1515/CRELLE.2006.049},
ZBLNUMBER = {1112.14022},
} -
[CurR] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, New York: Interscience Publishers, a division of John Wiley & Sons, 1962, vol. 11.
@book {CurR, MRKEY = {0144979},
AUTHOR = {Curtis, Charles W. and Reiner, Irving},
TITLE = {Representation Theory of Finite Groups and Associative Algebras},
SERIES = {Pure and Applied Math.},
VOLUME={11},
PUBLISHER = {Interscience Publishers, a division of John Wiley \& Sons},
ADDRESS={New York},
YEAR = {1962},
PAGES = {xiv+685},
MRCLASS = {20.80},
MRNUMBER = {0144979},
MRREVIEWER = {W. E. Jenner},
ZBLNUMBER = {0131.25601},
} -
[Eber] P. B. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago, IL: University of Chicago Press, 1996.
@book {Eber, MRKEY = {1441541},
AUTHOR = {Eberlein, Patrick B.},
TITLE = {Geometry of Nonpositively Curved Manifolds},
SERIES = {Chicago Lectures in Math,},
PUBLISHER = {University of Chicago Press},
ADDRESS = {Chicago, IL},
YEAR = {1996},
PAGES = {vii+449},
ISBN = {0-226-18197-9; 0-226-18198-7},
MRCLASS = {53-02 (53C20 53C21 53C35)},
MRNUMBER = {1441541},
MRREVIEWER = {Mar{\'ı}a J. Druetta},
ZBLNUMBER = {0883.53003},
} -
[EMO]
A. Eskin, S. Mozes, and H. Oh, "On uniform exponential growth for linear groups," Invent. Math., vol. 160, iss. 1, pp. 1-30, 2005.
@article {EMO, MRKEY = {2129706},
AUTHOR = {Eskin, Alex and Mozes, Shahar and Oh, Hee},
TITLE = {On uniform exponential growth for linear groups},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {160},
YEAR = {2005},
NUMBER = {1},
PAGES = {1--30},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {20F69 (20F05 20G20)},
MRNUMBER = {2129706},
MRREVIEWER = {Victor M. Petrogradsky},
DOI = {10.1007/s00222-004-0378-z},
ZBLNUMBER = {1137.20024},
} -
[FL]
C. Favre and J. Rivera-Letelier, "Équidistribution quantitative des points de petite hauteur sur la droite projective," Math. Ann., vol. 335, iss. 2, pp. 311-361, 2006.
@article {FL, MRKEY = {2221116},
AUTHOR = {Favre, Charles and Rivera-Letelier, Juan},
TITLE = {\'{E}quidistribution quantitative des points de petite hauteur sur la droite projective},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {335},
YEAR = {2006},
NUMBER = {2},
PAGES = {311--361},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {11G50 (37F10)},
MRNUMBER = {2221116},
MRREVIEWER = {Matthew H. Baker},
DOI = {10.1007/s00208-006-0751-x},
ZBLNUMBER = {1175.11029},
} -
[Gel]
T. Gelander, "Homotopy type and volume of locally symmetric manifolds," Duke Math. J., vol. 124, iss. 3, pp. 459-515, 2004.
@article {Gel, MRKEY = {2084613},
AUTHOR = {Gelander, Tsachik},
TITLE = {Homotopy type and volume of locally symmetric manifolds},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {124},
YEAR = {2004},
NUMBER = {3},
PAGES = {459--515},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {53C24 (22E40 53C20 57N16)},
MRNUMBER = {2084613},
MRREVIEWER = {Dave Witte Morris},
DOI = {10.1215/S0012-7094-04-12432-7},
ZBLNUMBER = {1076.53040},
} -
[IwaMat]
N. Iwahori and H. Matsumoto, "On some Bruhat decomposition and the structure of the Hecke rings of $\mathit{p}$-adic Chevalley groups," Inst. Hautes Études Sci. Publ. Math., iss. 25, pp. 5-48, 1965.
@article {IwaMat, MRKEY = {0185016},
AUTHOR = {Iwahori, N. and Matsumoto, H.},
TITLE = {On some {B}ruhat decomposition and the structure of the {H}ecke rings of {$\mathit{p}$}-adic {C}hevalley groups},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {25},
YEAR = {1965},
PAGES = {5--48},
ISSN = {0073-8301},
MRCLASS = {20.70 (14.50)},
MRNUMBER = {0185016},
MRREVIEWER = {Rimhak Ree},
URL = {http://www.numdam.org/item?id=PMIHES_1965__25__5_0},
ZBLNUMBER = {0228.20015},
} -
[Jab] N. Jacobson, Lie Algebras, New York: Interscience Publishers, a division of John Wiley & Sons, 1962, vol. 10.
@book {Jab, MRKEY = {0143793},
AUTHOR = {Jacobson, Nathan},
TITLE = {Lie Algebras},
SERIES = {Interscience Tracts in Pure and Applied Math.},
VOLUME={10},
PUBLISHER = {Interscience Publishers, a division of John Wiley \& Sons},
ADDRESS={New York},
YEAR = {1962},
PAGES = {ix+331},
MRCLASS = {17.30 (22.80)},
MRNUMBER = {0143793},
MRREVIEWER = {Rimhak Ree},
ZBLNUMBER = {0121.27504},
} -
[KM] D. A. Kazdan and G. A. Margulis, "A proof of Selberg’s hypothesis," Mat. Sb., vol. 75 (117), pp. 163-168, 1968.
@article {KM, MRKEY = {0223487},
AUTHOR = {Kazdan, D. A. and Margulis, G. A.},
TITLE = {A proof of {S}elberg's hypothesis},
JOURNAL = {Mat. Sb.},
VOLUME = {75 (117)},
YEAR = {1968},
PAGES = {163--168},
MRCLASS = {22.50},
MRNUMBER = {0223487},
MRREVIEWER = {A. Borel},
ZBLNUMBER={0241.22024},
} -
[Land]
E. Landvogt, "Some functorial properties of the Bruhat-Tits building," J. Reine Angew. Math., vol. 518, pp. 213-241, 2000.
@article {Land, MRKEY = {1739403},
AUTHOR = {Landvogt, Erasmus},
TITLE = {Some functorial properties of the {B}ruhat-{T}its building},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {518},
YEAR = {2000},
PAGES = {213--241},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {20E42 (22E50 51E24)},
MRNUMBER = {1739403},
MRREVIEWER = {Curtis D. Bennett},
DOI = {10.1515/crll.2000.006},
ZBLNUMBER = {0937.20026},
} -
[Lang0] S. Lang, Algebra, third ed., New York: Springer-Verlag, 2002, vol. 211.
@book {Lang0, MRKEY = {1878556},
AUTHOR = {Lang, Serge},
TITLE = {Algebra},
SERIES = {Graduate Texts in Math.},
VOLUME = {211},
EDITION = {third},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2002},
PAGES = {xvi+914},
ISBN = {0-387-95385-X},
MRCLASS = {00A05 (15-02)},
MRNUMBER = {1878556},
ZBLNUMBER = {0984.00001},
} -
[Lang] S. Lang, Fundamentals of Diophantine Geometry, New York: Springer-Verlag, 1983.
@book {Lang, MRKEY = {0715605},
AUTHOR = {Lang, Serge},
TITLE = {Fundamentals of {D}iophantine Geometry},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1983},
PAGES = {xviii+370},
ISBN = {0-387-90837-4},
MRCLASS = {11-02 (11Dxx 11Gxx 14G25)},
MRNUMBER = {0715605},
MRREVIEWER = {Gerd Faltings},
ZBLNUMBER = {0528.14013},
} -
[MW]
D. W. Masser and G. Wüstholz, "Fields of large transcendence degree generated by values of elliptic functions," Invent. Math., vol. 72, iss. 3, pp. 407-464, 1983.
@article {MW, MRKEY = {0704399},
AUTHOR = {Masser, D. W. and W{ü}stholz, G.},
TITLE = {Fields of large transcendence degree generated by values of elliptic functions},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {72},
YEAR = {1983},
NUMBER = {3},
PAGES = {407--464},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11J89 (11G35 14G25)},
MRNUMBER = {Masser},
MRREVIEWER = {Michel Waldschmidt},
DOI = {10.1007/BF01398396},
} -
@article {Mos, MRKEY = {0069830},
AUTHOR = {Mostow, G. D.},
TITLE = {Self-adjoint groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {62},
YEAR = {1955},
PAGES = {44--55},
ISSN = {0003-486X},
MRCLASS = {20.0X},
MRNUMBER = {0069830},
MRREVIEWER = {K. Iwasawa},
DOI = {10.2307/2007099},
ZBLNUMBER = {0065.01404},
} -
[OnVin] A. L. Onishchik and È. B. Vinberg, Lie Groups and Algebraic Groups, New York: Springer-Verlag, 1990.
@book {OnVin, MRKEY = {1064110},
AUTHOR = {Onishchik, A. L. and Vinberg, {È}. B.},
TITLE = {Lie Groups and Algebraic Groups},
SERIES = {Springer Ser. Soviet Math.},
NOTE = {translated from the Russian and with a preface by D. A. Leites},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1990},
PAGES = {xx+328},
ISBN = {3-540-50614-4},
MRCLASS = {22-01 (17B20 20G20 22E10 22E15)},
MRNUMBER = {1064110},
MRREVIEWER = {James E. Humphreys},
ZBLNUMBER = {0722.22004},
} -
[Pi]
J. Pineiro, L. Szpiro, and T. J. Tucker, "Mahler measure for dynamical systems on ${\Bbb P}^1$ and intersection theory on a singular arithmetic surface," in Geometric Methods in Algebra and Number Theory, Boston, MA: Birkhäuser, 2005, vol. 235, pp. 219-250.
@incollection {Pi, MRKEY = {2166086},
AUTHOR = {Pineiro, Jorge and Szpiro, Lucien and Tucker, Thomas J.},
TITLE = {Mahler measure for dynamical systems on {${\Bbb P}\sp 1$} and intersection theory on a singular arithmetic surface},
BOOKTITLE = {Geometric Methods in Algebra and Number Theory},
SERIES = {Progr. Math.},
VOLUME = {235},
PAGES = {219--250},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2005},
MRCLASS = {14G40 (11G50 11R04 37F10)},
MRNUMBER = {2166086},
MRREVIEWER = {Robin de Jong},
DOI = {10.1007/0-8176-4417-2_10},
ZBLNUMBER = {1101.11020},
} -
[Rag] M. S. Raghunathan, Discrete Subgroups of Lie Groups, New York: Springer-Verlag, 1972, vol. 68.
@book {Rag, MRKEY = {0507234},
AUTHOR = {Raghunathan, M. S.},
TITLE = {Discrete Subgroups of {L}ie Groups},
SERIES = {Ergeb. Mat. Grenz.},
VOLUME={68},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1972},
PAGES = {ix+227},
MRCLASS = {22E40},
MRNUMBER = {0507234},
MRREVIEWER = {J. S. Joel},
ZBLNUMBER = {0254.22005},
} -
[Sch]
A. Schinzel, Polynomials with Special Regard to Reducibility, Cambridge: Cambridge Univ. Press, 2000, vol. 77.
@book {Sch, MRKEY = {1770638},
AUTHOR = {Schinzel, A.},
TITLE = {Polynomials with Special Regard to Reducibility},
SERIES = {Encycl. Math. Appl.},
VOLUME = {77},
NOTE = {with an appendix by Umberto Zannier},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2000},
PAGES = {x+558},
ISBN = {0-521-66225-7},
MRCLASS = {11R09 (11C08 12D05 12E05 12E25)},
MRNUMBER = {1770638},
MRREVIEWER = {Art{ū}ras Dubickas},
DOI = {10.1017/CBO9780511542916},
ZBLNUMBER = {0956.12001},
} -
[Schur] I. Schur, Über Gruppen periodischer Substitutionen, Sitzber. Preuss. Akad. Wiss., 1911.
@book{Schur,
author={Schur, I.},
TITLE={Über Gruppen periodischer Substitutionen},
PUBLISHER={Sitzber. Preuss. Akad. Wiss.},
YEAR={1911},
JFMNUMBER={42.0155.01},
PAGES={619--627},
} -
[Sh]
Y. Shalom, "Explicit Kazhdan constants for representations of semisimple and arithmetic groups," Ann. Inst. Fourier $($Grenoble$)$, vol. 50, iss. 3, pp. 833-863, 2000.
@article {Sh, MRKEY = {1779896},
AUTHOR = {Shalom, Yehuda},
TITLE = {Explicit {K}azhdan constants for representations of semisimple and arithmetic groups},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {50},
YEAR = {2000},
NUMBER = {3},
PAGES = {833--863},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {22E46 (22D10 22D30 22E35 43A15)},
MRNUMBER = {1779896},
MRREVIEWER = {Paul Jolissaint},
URL = {http://www.numdam.org/item?id=AIF_2000__50_3_833_0},
ZBLNUMBER = {0966.22004},
} -
[Sm]
C. Smyth, "The Mahler measure of algebraic numbers: a survey," in Number Theory and Polynomials, Cambridge: Cambridge Univ. Press, 2008, vol. 352, pp. 322-349.
@incollection {Sm, MRKEY = {2428530},
AUTHOR = {Smyth, Chris},
TITLE = {The {M}ahler measure of algebraic numbers: a survey},
BOOKTITLE = {Number Theory and Polynomials},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {352},
PAGES = {322--349},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2008},
MRCLASS = {11R06},
MRNUMBER = {2428530},
DOI = {10.1017/CBO9780511721274.021},
} -
[SUZ]
L. Szpiro, E. Ullmo, and S. Zhang, "Équirépartition des petits points," Invent. Math., vol. 127, iss. 2, pp. 337-347, 1997.
@article {SUZ, MRKEY = {1427622},
AUTHOR = {Szpiro, L. and Ullmo, E. and Zhang, S.},
TITLE = {\'{E}quirépartition des petits points},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {127},
YEAR = {1997},
NUMBER = {2},
PAGES = {337--347},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14G40 (11G35 14G05)},
MRNUMBER = {1427622},
MRREVIEWER = {Antoine Chambert-Loir},
DOI = {10.1007/s002220050123},
ZBLNUMBER = {0991.11035},
} -
[Stein] R. Steinberg, Lectures on Chevalley Groups, New Haven, Conn.: Yale University, 1968.
@book {Stein, MRKEY = {0466335},
AUTHOR = {Steinberg, Robert},
TITLE = {Lectures on {C}hevalley Groups},
NOTE = {notes prepared by John Faulkner and Robert Wilson},
PUBLISHER = {Yale University},
ADDRESS={New Haven, Conn.},
YEAR = {1968},
PAGES = {iii+277},
MRCLASS = {20G15 (14LXX)},
MRNUMBER = {0466335},
MRREVIEWER = {J. E. Humphreys},
ZBLNUMBER = {1196.22001},
} -
[Tala]
V. Talamanca, "A Gelfand-Beurling type formula for heights on endomorphism rings," J. Number Theory, vol. 83, iss. 1, pp. 91-105, 2000.
@article {Tala, MRKEY = {1767654},
AUTHOR = {Talamanca, Valerio},
TITLE = {A {G}elfand-{B}eurling type formula for heights on endomorphism rings},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {83},
YEAR = {2000},
NUMBER = {1},
PAGES = {91--105},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11G50 (11R54)},
MRNUMBER = {1767654},
MRREVIEWER = {Jeffrey Lin Thunder},
DOI = {10.1006/jnth.1999.2506},
ZBLNUMBER = {0965.16018},
} -
@article {Tits, MRKEY = {0286898},
AUTHOR = {Tits, J.},
TITLE = {Free subgroups in linear groups},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {20},
YEAR = {1972},
PAGES = {250--270},
ISSN = {0021-8693},
MRCLASS = {20.75},
MRNUMBER = {0286898},
MRREVIEWER = {B. A. F. Wehrfritz},
DOI = {10.1016/0021-8693(72)90058-0},
ZBLNUMBER = {0236.20032},
} -
[Thu] W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton, NJ: Princeton Univ. Press, 1997, vol. 35.
@book {Thu, MRKEY = {1435975},
AUTHOR = {Thurston, William P.},
TITLE = {Three-Dimensional Geometry and Topology. {V}ol. 1},
SERIES = {Princeton Math. Ser.},
VOLUME = {35},
NOTE = {edited by Silvio Levy},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1997},
PAGES = {x+311},
ISBN = {0-691-08304-5},
MRCLASS = {57M50 (53A35 57M25 57M60 57N10)},
MRNUMBER = {1435975},
MRREVIEWER = {Athanase Papadopoulos},
ZBLNUMBER = {0873.57001},
} -
[Ull]
E. Ullmo, "Positivité et discrétion des points algébriques des courbes," Ann. of Math., vol. 147, iss. 1, pp. 167-179, 1998.
@article {Ull, MRKEY = {1609514},
AUTHOR = {Ullmo, Emmanuel},
TITLE = {Positivité et discrétion des points algébriques des courbes},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {147},
YEAR = {1998},
NUMBER = {1},
PAGES = {167--179},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {14G40 (11G30 14H25 14H40)},
MRNUMBER = {1609514},
MRREVIEWER = {Yuichiro Takeda},
DOI = {10.2307/120987},
ZBLNUMBER = {0934.14013},
} -
[Wang] H. C. Wang, "Topics on totally discontinuous groups," in Symmetric Spaces, New York: Dekker, 1972, vol. 8, pp. 459-487.
@incollection {Wang, MRKEY = {0414787},
AUTHOR = {Wang, Hsien Chung},
TITLE = {Topics on totally discontinuous groups},
BOOKTITLE = {Symmetric Spaces},
VENUE={{S}hort {C}ourses, {W}ashington {U}niv., {S}t. {L}ouis, {M}o., 1969--1970},
PAGES = {459--487},
SERIES={Pure and Appl. Math.},
VOLUME={8},
PUBLISHER = {Dekker},
ADDRESS = {New York},
YEAR = {1972},
MRCLASS = {22E40},
MRNUMBER = {0414787},
MRREVIEWER = {A. L. Oniscik},
ZBLNUMBER = {0232.22018},
} -
[Werf] B. A. F. Wehrfritz, Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices, New York: Springer-Verlag, 1973, vol. 76.
@book {Werf, MRKEY = {0335656},
AUTHOR = {Wehrfritz, B. A. F.},
TITLE = {Infinite Linear Groups. {A}n Account of the Group-Theoretic Properties of Infinite Groups of Matrices},
SERIES={Ergeb. Mat. Grenz.},
VOLUME={76},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1973},
PAGES = {xiv+229},
MRCLASS = {20HXX},
MRNUMBER = {0335656},
MRREVIEWER = {I. Stewart},
ZBLNUMBER = {0261.20038},
} -
[Zhang] S. Zhang, "Small points and adelic metrics," J. Algebraic Geom., vol. 4, iss. 2, pp. 281-300, 1995.
@article {Zhang, MRKEY = {1311351},
AUTHOR = {Zhang, Shou-Wu},
TITLE = {Small points and adelic metrics},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {4},
YEAR = {1995},
NUMBER = {2},
PAGES = {281--300},
ISSN = {1056-3911},
MRCLASS = {14G40},
MRNUMBER = {1311351},
MRREVIEWER = {Antoine Chambert-Loir},
ZBLNUMBER = {0861.14019},
} -
[Zhang0]
S. Zhang, "Equidistribution of small points on abelian varieties," Ann. of Math., vol. 147, iss. 1, pp. 159-165, 1998.
@article {Zhang0, MRKEY = {1609518},
AUTHOR = {Zhang, Shou-Wu},
TITLE = {Equidistribution of small points on abelian varieties},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {147},
YEAR = {1998},
NUMBER = {1},
PAGES = {159--165},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {14G40 (11G10 14G05 14K15)},
MRNUMBER = {1609518},
MRREVIEWER = {Yuichiro Takeda},
DOI = {10.2307/120986},
ZBLNUMBER = {0991.11034},
}