Alan Reid, Edgar Odell Lovett Professor of Mathematics at Rice University, Mathematics Department Chair
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Publications

  1. (With C. Maclachlan) Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups. Math. Proc. Camb. Phil. Soc. 102 (1987), 251–257.
  2. (With C. Maclachlan) The arithmetic structure of tetrahedral groups of hyperbolic isometries. Mathematika 36 (1990), 221–240.
  3. A note on trace-fields of Kleinian groups. Bull. London Math. Soc. 22 (1990), 349–352.
  4. Arithmeticity of knot complements. J. London Math. Soc. 43 (1991), 171–184.
  5. Totally geodesic surfaces in hyperbolic 3-manifolds. Proc. Edinburgh Math. Soc. 34 (1991), 77–88.
  6. (With W. D. Neumann) Amalgamation and the invariant trace-field of Kleinian groups. Math. Proc. Camb. Phil. Soc. 109 (1991), 509–515.
  7. (With C. Maclachlan) Parameterizing Fuchsian subgroups of Bianchi groups. Canadian J. Math. 43 (1991), 158–181.
  8. Commensurability and isospectrality of arithmetic hyperbolic 2 and 3-manifolds. Duke Math J. 65 (1992), 215–228.
  9. (With W. D. Neumann) Arithmetic of hyperbolic manifolds. In TOPOLOGY '90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University. De Gruyter Verlag (1992), 273–310.
  10. (With W. D. Neumann) Notes on Adams' small volume orbifolds. In TOPOLOGY '90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University. De Gruyter Verlag (1992), 311–314.

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  11. (With W. Menasco) Totally geodesic surfaces in hyperbolic link complements. In TOPOLOGY '90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University. De Gruyter Verlag (1992), 215–226.
  12. Some remarks on 2-generator hyperbolic 3-manifolds. In Discrete Groups and Geometry, Proceedings in honour of the retirement of A. M. Macbeath. L. M. S. Lecture Note Series 173 (1992), 209–219.
  13. (With W. D. Neumann) Rigidity of cusps in deformations of hyperbolic 3-orbifolds. Math. Ann. 295 (1993), 223-237.
  14. (With D. D. Long) Closed incompressible surfaces in 2-generator hyperbolic 3-manifolds with a single cusp. Proc. Edinburgh Math. Soc. 36 (1993), 501–513.
  15. (With T. Chinburg) Closed hyperbolic 3-manifolds whose closed geodesics all are simple. J. Diff. Geom. 38 (1993), 545–558.
  16. (With C. Adams) Quasi-Fuchsian surfaces in hyperbolic knot complements. Journal Australian Math. Soc. 55 (1993), 116–131.
  17. (With K. N. Jones) Non-simple closed geodesics in hyperbolic 3-manifolds. Math. Proc. Camb. Phil. Soc. 116 (1994), 339–351.
  18. (With D. Cooper and D. D. Long) Bundles and finite foliations. Invent. Math. 118 (1994), 255–283.
  19. (With B. Bowditch and C. Maclachlan) Arithmetic hyperbolic surface bundles. Math. Annalen 302 (1995), 31–60.
  20. A non-Haken hyperbolic 3-manifold covered by a surface bundle.Pacific J. Math. 167 (1995), 163–182.

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  21. (With C. McA. Gordon) Tangle decompositions of tunnel number one knots and links. Journal of Knot Theory and its Ramifications 4 (1995), 389–409.
  22. (With C. Adams) Unknotting tunnels in two-bridge knot and link complements. Comment. Math. Helv. 71 (1996), 617–627.
  23. (With D. D. Long and C. Maclachlan) Splitting groups of signature (1;n). J. Algebra 185 (1996), 329–341. (pdf)
  24. (With J. W. Anderson) Commensurability of Fuchsian groups and their axes. Proc. A. M. S. 125 (1997), 941–942. (pdf)
  25. (With D. Cooper and D. D. Long) Finite foliations and similarity interval exchange maps. Topology 36 (1997), 209–227. (pdf)
  26. (With D. Cooper and D. D. Long) Essential closed surfaces in bounded 3-manifolds. Journal of The American Math. Soc. 10, (1997), 553–563. (pdf)
  27. (With F. W. Gehring, C. Maclachlan and G. J. Martin) Arithmeticity, Discreteness and Volume. Trans. A. M. S. 349 (1997), 3611–3643. (pdf)
  28. (With C. Maclachlan) Generalised Fibonacci Manifolds. J. of Transformation Groups, 2 (1997), 165–182.
  29. (With K. N. Jones) Geodesic intersections in arithmetic hyperbolic 3-manifolds. Duke Math. J. 89 (1997), 75–86.
  30. (With C. Maclachlan) Invariant trace-fields and quaternion algebras of polyhedral groups. J. London Math. Soc. 58 (1998), 709–722. (pdf)

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  31. (With D. D. Long) Simple quotients of hyperbolic 3-manifold groups. Proc. A. M. S. 126 (1998), 877–880. (pdf)
  32. (With D. D. Long) On Fuchsian groups with the same set of axes. Bull. London Math. Soc. 30 (1998), 533 – 538. (pdf)
  33. (With D. D. Long) Free products with amalgamation and p-adic Lie Groups, Canadian Math. Bulletin, 41 (1998), 423–433.
  34. (With D. Cooper and D. D. Long) Infinite Coxeter groups are virtually indicable. Proc. Edinburgh Math. Soc. 41 (1998), 303–313.
  35. (With K. N. Jones) Minimal index torsion-free subgroups in Kleinian groups. Math. Annalen 310 (1998), 235–250. (pdf)
  36. (With P. Callahan) Hyperbolic structures on knot complements. Chaos, Solitons and Fractals 9 (1998), 705–738. (pdf)
  37. (With S. Wang) Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree. Communications in Analysis and Geometry, 7 (1999) 105–132
  38. (With D. D. Long) Commensurability and the character variety, Math. Research Letters 6 (1999), 581–591. (pdf)
  39. (With D. D. Long) On the geometric boundaries of hyperbolic 4-manifolds, Geometry and Topology, 4 (2000), 171–178. (pdf)
  40. (With C. Adams) Systoles of hyperbolic 3-manifolds. Math. Proc. Cambridge Philos. Soc. 128 (2000), 103–110. (pdf)

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  41. (With K. N. Jones) Vol3 and other exceptional hyperbolic 3-manifolds, Proc. Amer. Math. Soc. 129 (2001), 2175–2185. (pdf)
  42. (With D. D. Long) The fundamental group of the double of the figure eight knot exterior is GFERF, Bull. London Math. Soc. 33 (2001), 391–396.
  43. (With D. D. Long) Constructing hyperbolic manifolds which bound geometrically, Math. Research Letters 8 (2001), 443–456. (pdf)
  44. (With D. D. Long) On subgroup separability in hyperbolic Coxeter groups, Geometriae Dedicata, 87 (2001), 245–260. (pdf)
  45. (With I. Agol and D. D. Long) The Bianchi groups are separable on geometrically finite subgroups, Annals of Math. 153 (2001), 599–621. (pdf)
  46. (With T. Chinburg, E. Friedman and K. N. Jones) The smallest volume arithmetic hyperbolic 3-manifold, Annali della Scuola Normale Superiore di Pisa 30 (2001), 1–40.
  47. (With D. D. Long) All flat manifolds are cusps of hyperbolic orbifolds, Algebraic and Geometric Topology 2 (2002), 285–296. (pdf)
  48. (With D. D. Long) Pseudomodular surfaces, Journal für de Reine Angew. 552 (2002), 77–100. (pdf)
  49. (With S. Wang and Q. Zhou) Generalized Hopfian property, a minimal Haken manifold and epimorphisms between 3-manifold groups, Acta Mathematica Sinica, English Series 18 (2002), 157–172. (pdf)
  50. (With C. Leininger) The co-rank conjecture for 3-manifold groups, Algebraic and Geometric Topology 2 (2002), 37–50. (pdf)

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  51. (With M. D. Baker) Arithmetic knots in closed 3-manifolds, in Proceedings of Knots 2000, J. Knot Theory and its Ramifications 11 (2002), 903–920. (pdf)
  52. (With D. D. Long) Integral points on character varieties, Math. Annalen. 325 (2003), 299–321. (pdf)
  53. (With E. Hamilton) Eigenvalue fields of hyperbolic orbifolds, Proc. A. M. S. 132 (2004), 2497–2503. (pdf)
  54. (With C. McA. Gordon and D. D. Long) Surface subgroups of Coxeter and Artin groups, J. Pure and Applied Algebra 189 (2004), 135–148. (pdf)
  55. (With D. D. Long) Generalized Dedekind sums, Geometry and Topology Monographs 7, Proceedings of the Casson Fest (2004), 205–212. (pdf)
  56. (With D. D. Long) On asymmetric hyperbolic manifolds, Math. Proc. Camb. Phil. Soc.138 (2005), 301–306. (pdf)
  57. (With C. Leininger) A combination theorem for Veech subgroups of the Mapping class group, GAFA 16 (2006), 403–436. (pdf)
  58. Surface subgroups of Mapping Class groups, in Problems about Mapping Class Groups, Proc. Symp. Pure and Applied Math., ed. B. Farb. (2006), 274–286. (pdf)
  59. (With D. D. Long and C. Maclachlan) Arithmetic Fuchsian groups of genus zero. Pure and Applied Math Quarterly 2 (2006), a special issue to celebrate the 60th birthday of Professor J. H. Coates, 569–599. (pdf)
  60. (With D. Cooper and D. D. Long) On the virtual Betti numbers of arithmetic hyperbolic 3–manifolds, Geometry and Topology 11 (2007) 2265–2276. (pdf)

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  61. The geometry and topology of arithmetic hyperbolic 3-manifolds, Proc. Symposium Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces, Kyoto 2006, RIMS Kokyuroku Series, 1571 (2007), 31–58. (pdf)
  62. (With D. D. Long and A. Lubotzky), Heegaard genus and Property for hyperbolic 3-manifolds, Journal of Topology 1 (2008) 152–158. (pdf)
  63. (With D. D. Long) Subgroup separability and virtual retractions of groups, Topology 47 (2008), 137–159. (pdf)
  64. (With D. D. Long) Finding fibre faces in finite covers, Math Research Letters 15 (2008), 521–524. (pdf)
  65. (With G. S. Walsh) Commensurability classes of 2-bridge knot complements, Algebraic and Geometric Topology 8 (2008), 1031–1057(pdf)
  66. (With T. Chinburg, E. Hamilton and D. D. Long) Geodesics and
    commensurability classes of arithmetic hyperbolic 3-manifolds, Duke Math. J. 145 (2008), 25–44.(pdf)
  67. (With C. Leininger, D. B. McReynolds and W. D. Neumann) Length and eigenvalue equivalence, International Math. Research Notices, volume 2007, article ID rnm135, 24 pages, doi:10.1093/imrn/rnm135. (pdf)
  68. (With M. Lackenby and D. D. Long) Covering spaces of arithmetic 3-orbifolds, International Math. Research Notices, volume 2008, article ID rnn036, 38 pages, doi: 10.1093/imrn/rnn036. (pdf)
  69. (With M. Lackenby and D. D. Long) LERF and the Lubotzky-Sarnak Conjecture, Geometry and Topology 12 (2008), 2047–2056(pdf)
  70. (With T. Chinburg and D. D. Long) Cusps of minimal non-compact arithmetic hyperbolic 3-orbifolds, Pure and Applied Math Quarterly 4 (2008), 1013–1031. A special issue to celebrate the 80th birthday of Professor J-P. Serre. (pdf)

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  71. (With P. Bonfert-Taylor, G. Martin and E. Taylor) Teichmüller mappings, quasi-conformal homogeneity, and non-amenable covers of Riemann surfaces, Pure Appl. Math. Q. 7 (2011), 455–468. A special issue to celebrate the 80th birthday of F. W. Gehring. (pdf)
  72. (With D. D. Long), Eigenvalues of hyperbolic elements in Kleinian
    groups, In the tradition of Ahlfors–Bers V, Contemp. Math. 510, 197–208, A.M.S. Publications, (2010). (pdf)
  73. (With M. Boileau, S. Boyer and S. Wang) Simon's Conjecture for 2-bridge knots in Communications in Analysis and Geometry 18 (2010), 121–143. (pdf)
  74. (With D. D. Long), Grothendieck's problem for 3-manifold groups in Groups, Geometry and Dynamics, 5 (2011), 479–499. (pdf)
  75. (With D. D. Long and M. Thistlethwaite), Zariski dense surface subgroups in SL(3,Z), Geometry and Topology 15 (2011), 1–9. (pdf)
  76. (With C. J. Leininger and D. D. Long) Commensurators of finitely generated non-free Kleinian groups, Algebraic and Geometric Topology, 11 (2011), 605–624. (pdf)
  77. (With D. D. Long), Fields of definition of canonical curves, Interactions between hyperbolic geometry, quantum topology and number theory, Contemporary Math. 541, 247–257, A.M.S. Publications, (2011). (pdf)
  78. (With D. D. Long), Small subgroups of SL(3,Z), Experimental Math. 20 (2011), 412–425. (pdf)
  79. (With G. Masbaum) All finite groups are involved in Mapping Class groups, Geometry and Topology 12 (2012), 1393–1411. (pdf)
  80. (With M. D. Baker) Restricting the topology of 1–cusped arithmetic 3–manifolds, Algebraic & Geometric Topology 13 (2013) 1273–1298. (pdf)

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  81. (With D. D. Long) Constructing thin subgroups of SL(4, R). International Math. Research Notices, volume 2013, article ID rns281, 11 pages,
    doi:10.1093/imrn/rns281(pdf)
  82. (With D. D. Long) Constructing thin groups, MSRI Proceedings of the Hot Topics Workshop Thin Groups and Super-strong Approximation 151-166. (pdf)
  83. (With D. B. McReynolds and M. Stover) Collisions at infinity in hyperbolic manifolds, Math. Proc. Camb. Phil. Soc. 155 (2013), 459–463 (pdf)
  84. (With D. B. McReynolds) The genus spectrum of hyperbolic 3-manifolds, Math. Research Letters 21 (2014), 169–185. (pdf)
  85. (With M. D. Baker) Principal congruence link complements, Ann. Fac. Sci. Toulouse 23 (2014), 1063–1092(pdf)
  86. Traces, lengths, axes and commensurability, Ann. Fac. Sci. Toulouse 23 (2014), 1103–1118. (pdf)
  87. (With M. R. Bridson) Nilpotent completions of groups, Grothendieck pairs and four problems of Baumslag, Int. Math. Research Notices Volume 2015, 2111–2140, doi: 10.1093/imrn/rnt353 (pdf)
  88. Profinite properties of discrete groups, Proceedings
    of Groups St Andrews 2013, L.M.S. Lecture Note Series  242 73–104,
    Cambridge Univ. Press (2015). (pdf)
  89. (With G. Masbaum) Frattini and related subgroups of Mapping Class Groups), Proceedings of the Steklov Institute, 292 (2016), a special issue to celebrate the 75th birthday of V. P. Platonov, 143–152. (pdf)
  90. (With M.R. Bridson and M. Conder), Determining Fuchsian groups by their finite quotients, Israel J. Math, 214 (2016), 1–41. (pdf)

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  91. (With D. D. Long) Thin surface subgroups in cocompact lattices in
    SL(3,R), Illinois J. Math. 60 (2016), 39–53, a special issue to celebrate the work of Wolfgang Haken. (pdf)
  92. (With S. Garoufalidis) Constructing 1-cusped isospectral non-isometric
    hyperbolic 3-manifolds, Journal of Topology and Analysis 10 (2018), 1–25.(pdf)
  93. (With M. D. Baker) Principal Congruence Links: Class number greater than 1, Experimental Math, 27 (2018), 395–413. (pdf)
  94. (With M. R. Bridson and H. Wilton) Profinite rigidty and surface bundles over
    the circle. Bull. London Math. Soc. 49 (2017) 831–841 (pdf)
  95. Profinite rigidity, The Proceedings of the 2018 I.C.M. Rio De Janeiro, Vol. 2 1211–1234. (pdf)
  96. (With A. Kolpakov and L. Slavich) Embedding arithmetic hyperbolic manifolds, Math. Research Letters 25 (2018), 1305–1328. (pdf)
  97. (With M. D. Baker) Congruence link complements–a 3-dimensional Rademacher Conjecture  in Cohomology of Arithmetic Groups, Proceedings of the 66th Birthday Conference of Joachim Schwermer, 177–200, (2018). (pdf)
  98. (With D. B. McReynolds) Determining hyperbolic 3-manifolds by their
    surfaces, Proc A.M.S 147 (2019), 443–450. (pdf)
  99. (With C. J. Leininger) Pseudo-Anosov homeomorphisms not arising from branched covers, Groups Geometry and Dynamics 14 (2020), 151–175. (pdf)
  100. (With D. D. Long) Sequences of high rank lattices of large systole containing a fixed genus surface group, New York  J. Math 25 (2019), 145–155. (pdf)

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  101. (With M. D. Baker and M. Goerner) All principal congruence link complements, J. Algebra 528 (2019), 497–504. (pdf)
    • Technical Report: All  principal congruence link groups (pdf)
    • All known principal congruence links (pdf)
  102. (With A. Kolpakov and S. Riolo) Many cusped hyperbolic 3-manifolds do not bound geometrically. Proc. A. M. S.148 (2020), 2233-2243 (pdf)
  103. (With M. R. Bridson) Profinite rigidity, fibering and the figure-eight knot, in What's Next?: The Mathematical Legacy of William P. Thurston, Annals of Math. Study 205, PUP (2020), 45-64. (pdf)
  104. (With A. Kontorovich, D. D. Long and A. Lubotzky) WHAT IS... a Thin Group? Notices of  the A.M.S. 66 (2019), 905–910. (pdf)
  105. (With D. D. Long) Virtually spinning hyperbolic manifolds,Virtually spinning hyperbolic manifolds, Proc. Edinburgh Math. Soc.63 (2020), 305--313. (pdf)
  106. (With K. L. Petersen) Gonality and genus of canonical components of character varieties, in Characters in low-dimensional topology, 263-292, Contemp. Math., 760, Amer. Math. Soc., 2020. (pdf)
  107. (With M. R. Bridson, D. B. McReynolds and R. Spitler) Absolute profinite rigidity and hyperbolic geometry, Annals of Math. 192 (2020), 679–719. (pdf)
    • Supplemental Magma code for Absolute profinite rigidity and hyperbolic 
      geometry. (pdf)
  108. (With T. Chinburg and M. Stover) Azumaya algebras and canonical 
    components. Int. Math. Research Notices. (pdf)
  109. (With J. Jung) Embedding closed totally geodesic surfaces in Bianchi orbifolds, to appear Math. Research Letters. (pdf) 
  110. (With M. Chu) Embedding closed hyperbolic 3-manifolds in small 
    volume hyperbolic 4-manifolds, Algebraic and Geometric Topology 21 (2021) 2627–2647. (pdf)

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  111. (With N. M. Rouse) Infinitely many knots with non-integral trace, Experimental Math.32 (2023), 514–526.(pdf)
  112. (With M. R. Bridson, D. B. McReynolds and R. Spitler) On the profinite rigidity of triangle groups, Bull. L.M.S. 53 (2021), 1849–1862. (pdf)
  113. (With M. R. Bridson) Profinite rigidity, Kleinian groups and the 
    cofinite Hopf property, Michigan Math. J. 72 (2022), in honor of Gopal Prasad, 25–49. (pdf)
  114. (With C. J. Leininger) Filling links in 3-manifolds of rank 2. 
    Appendix to: Filling links and spines in 3-manifolds, M. Freedman and 
    V. Krushkal, Communications in Analysis and Geometry, 31 (2023), 2307–2333. (pdf)
  115. (With M. D. Baker)Infinitely many arithmetic alternating links, 
    Algebraic and Geometric Topology 23 (2023), 2857–2866. (pdf)
  116. (With C. Maclachlan and A. Salgueiro) Some remarks on group actions on hyperbolic 3-manifolds, Art Discrete Appl. Math. 5 (2022), special issue in honor of Marston Conder's 65th birthday. (pdf)
  117. (With D. D. Long and M. Wolff) Most Hitchin representations are strongly dense, to appear Michigan J. Math. (pdf)
  118. (With L. Ferrari and A. Kolpakov) Infinitely many arithmetic 
    hyperbolic rational homology 3-spheres that bound geometrically. Trans A.M.S. 376 (2023), 1979–1997(pdf)
  119. (With T. Cheetham-West, A. Lubotzky and R. Spitler) Property FA is not
    a profinite property, to appear Groups, Geometry and Dynamics. (pdf)
  120. (With M.Chu) Totally geodesic hyperbolic 3-manifolds in hyperbolic 
    link complements of tori in S^4. Pacific J. Math. (325), 191–207(pdf)

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  121. (With C. Sell) Hyperbolic manifolds without spinC structures and non-vanishing higher order Stiefel-Whitney classes,  to appear Proc. A. 
    M. S. (pdf)
  122. (With D. D. Long) Strongly dense representations of surface groups, to appear Proc. Edinburgh Math. Soc.(pdf)
  123. (With M. D. Baker) Infinitely many arithmetic alternating links: Class 
    number greater than one, J. Knot Theory and its Ramifications 32.
    (pdf)

Preprints

  • (With M. Freedman, D. Kasprowski, M. Kreck and P. Teichner) 3-manifolds and integral lifts. Appendix to: Immersions of punctured 4-Manifolds. (pdf)
  • (With B. Martelli) The Dirac operator on cusped hyperbolic manifolds. (pdf) 
  • (With M. R. Bridson and R. Spitler) Absolute profinite rigidity, 
    direct products, and finite presentability. (pdf)

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In Progress

  • (With C. McA. Gordon and J. Nogueira) Planar surfaces and meridional  
    subgroups of knot groups.

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Books

In the Tradition of Ahlfors-Bers VII,  Contemporary Math. 696 American Math. Soc. (2017). Co-editors: A. Basmajian and Y. Minsky.

In the Tradition of Ahlfors-Bers VI, Contemporary Mathematics 590 American Mathematical Society (2013). Co-editors: U. Hamenstadt, R. Rodriguez, S. Rhode, and M. Wolf. (pdf)

Surface Subgroups and Subgroup Separability in 3-Manifold Topology, With D. D. Long, Publicações Matemáticas no IMPA, (2005) (pdf)

The Arithmetic of Hyperbolic 3-Manifolds, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)

TOPOLOGY '90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University. Co-Editors: B. Apanasov, W. D. Neumann and L. Siebenmann, De Gruyter Verlag, (1992)

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Recent Lectures

Three Lectures at the Workshop on Geometric topology in Cortona
June 4–10, 2017 at the Palazzone in Cortona. (pdf)

Lectures at 2018 NZMRI Summer Workshop in Algebra and Representation Theory, Nelson NZ, January 7–13, 2018.
Click here and scroll down for slides.

Profinite rigidity, 2018 ICM talk, Rio De Janeiro, August 2018. (pdf)

Swiss Mathematical Society Annual Meeting, Public Lecture, June 2019 (pdf)

The Vinberg Distinguished Lecture series, October 2021 (virtual) - (opens in a new window)

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