New non-arithmetic lattices in SU(2,1) J. Paupert (Arizona State University), joint with M. Deraux (Grenoble), J.R. Parker (Durham). ICERM, september 2013 Introduction: lattices in real semisimple Lie groups Main results Complex Hyperbolic Space and Isometries Mostow’s lattices Configuration space of symmetric complex reflection triangle groups Sporadic groups Discreteness and Fundamental Domains Commensurability classes Non-arithmeticity Appendix: statement of Poincar´e polyhedron theorem Γ < G is a lattice if Γ\G has finite volume (Haar measure). Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Question 1: Do there exist lattices in G? Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis). G: real semisimple Lie group. Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Question 1: Do there exist lattices in G? Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis). G: real semisimple Lie group. Γ < G is a lattice if Γ\G has finite volume (Haar measure). Question 1: Do there exist lattices in G? Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis). G: real semisimple Lie group. Γ < G is a lattice if Γ\G has finite volume (Haar measure). Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis). G: real semisimple Lie group. Γ < G is a lattice if Γ\G has finite volume (Haar measure). Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Question 1: Do there exist lattices in G? Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis). G: real semisimple Lie group. Γ < G is a lattice if Γ\G has finite volume (Haar measure). Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Question 1: Do there exist lattices in G? Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis). G: real semisimple Lie group. Γ < G is a lattice if Γ\G has finite volume (Haar measure). Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Question 1: Do there exist lattices in G? Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis). G: real semisimple Lie group. Γ < G is a lattice if Γ\G has finite volume (Haar measure). Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Question 1: Do there exist lattices in G? Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). G: real semisimple Lie group. Γ < G is a lattice if Γ\G has finite volume (Haar measure). Γ is cocompact (C) if Γ\G is compact, non-cocompact (NC) otherwise. Question 1: Do there exist lattices in G? Yes (Borel–Harish-Chandra), the so-called arithmetic lattices (e.g. SL(n,Z) < SL(n,R), SL(n,Z[i]) or SL(n,Z[ω]) < SL(n,C)). Infinitely many, C and NC. Question 2: Do there exist any other lattices in G? If yes, how many? (cid:73) In SL(2,R): yes, lots. (Arithmetic=small, Teichmu¨ller=BIG). (cid:73) No, if R-Rank(G) (cid:62) 2 (Margulis).