Introduction to Arithmetic Groups What Is an Arithmetic Group? Geometric

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What is an arithmetic group? Introduction to Arithmetic Groups Every arithmetic group is a group of matrices with integer entries. Dave Witte Morris More precisely, SLΓ(n, Z) G : GZ where " ∩ = University of Lethbridge, Alberta, Canada ai,j Z, http://people.uleth.ca/ dave.morris SL(n, Z) nΓ n mats (aij) ∈ ∼ = " × # det 1 $ [email protected] # = # semisimple G SL(n, R) is connected Lie# group , Abstract. This lecture is intended to introduce ⊆ # def’d over Q non-experts to this beautiful topic. Examples For further reading, see: G SL(n, R) SL(n, Z) = ⇒ = 2 2 2 D. W. Morris, Introduction to Arithmetic Groups. G SO(1, n) Isom(x1 x2 xn 1) = = Γ − − ···− + Deductive Press, 2015. SO(1, n)Z ⇒ = http://arxiv.org/src/math/0106063/anc/ subgroup of that has finite index Γ Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 1 / 12 Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 2 / 12 Γ Geometric motivation Other spaces yield groups that are more interesting. Group theory = the study of symmetry Rn is a symmetric space: y Example homogeneous: x Symmetries of a tessellation (periodic tiling) every pt looks like all other pts. x,y, isometry x ! y. 0 ∀ ∃ x0 reflection through a point y0 (x+ x) is an isometry. = − Assume tiles of tess of X are compact (or finite vol). Then symmetry group is a lattice in Isom(X) G: = symmetry group Z2 Z2 G/ is compact (or has finite volume). = " is cocompact Γis noncocompact Thm (Bieberbach, 1910). tess of Rn, Zn. or uniform or nonuniform Γ ∀ Γ " Γ Other spaces yield groups that are more interesting. ManyΓ lattices in G areΓ arithmetic subgroups. Γ Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 3 / 12 Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 4 / 12 2 Eg. Tess’ns of hyperbolic plane h . (symmetric space) Other interesting groups G: simple Lie group. G SO(1, 2) SL(2, R). m 2 n 2 " KaleidoTile" SO(m, n) Isom k 1 xk k 1 xm k = = − = + (lattice: GZ) % & & ' 2 SU(m, n): change R to C and xk to xk xk (lattice: GZ Zi) + Sp(m, n): change C to H

(lattice: GZ Zi Zj Zk GHZ ) + + + = SL(n, R), SL(n, C), SL(n, H) Sp(2n, R), Sp(2n, C), SO(n, H) c Wikipedia , = cocompact lattice SL(2, Z) SO(1, 2)Z ﬁnitely many “exceptional grps” (E6,E7,E8,F4,G2) = " Tessellations of other symmetric spaces Γ Γ Semisimple Lie group G1 G2 Gr . correspond to lattices in other interesting groups. " × × ···

Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 5 / 12 Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 6 / 12

Arithmetic groups are lattices There are many arithmetic lattices in m 2 n 2 SO(m, n) Isom k 1 xk k 1 xm k . = = − = + Tessellations of other symmetric spaces %& & ' correspond to lattices in semisimple Lie groups. Proof. Choose a1,a2,...,am n Q+. m 2 n+ ∈ 2 Let Q(x) k 1 ak xk k 1 am k xm k. = = − = + + Theorem (Borel and Harish-Chandra) R change of& vars takes Q(x)& to the standard form. Every arithmetic group is a lattice: Therefore Isom Q(x) SO(m, n), - Semisimple Lie group G deﬁned over Q GZ is a lattice. so Isom Q(x) " latt in SO(m, n). ⇒ % Z ' % ' Remark Fact. Isom Q(x) Z nonuniform There are many arithmetic lattices in : m n G # Q(x)% 0'has nonzero soln in Z + . Many ways to embed G in SL(n, R) (if n is large) = many diﬀerent possibilities for GZ. Eg. Isom x2 x2 x2 x2 is noncocompact latt, ⇒ 1 − 2 − 3 − 3 Z Some are cocpct, and some are not cocpct [Borel]. but Isom 7x2 x2 x2 x2 is cocompact latt. % 1 − 2 − 3 − '3 Z Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 7 / 12 Dave Witte Morris (U% of Lethbridge) Introduction to Arithmetic' Groups Auckland, Feb 2020 8 / 12 Theorem (Borel and Harish-Chandra) Basic algebraic properties Every arithmetic group is a lattice. Proposition

Converse: GZ is finitely generated. Margulis Arithmeticity Theorem Proof when is cocompact. Every lattice in simple G is an arithmetic subgroup Choose bounded, open C G, such that GZ C G. unless G SO(1, n) or SU(1, n). ⊂ = Let S s ΓGZ sC C ≠ . " = { ∈ | ∩ ∅ } S is finite, because GZ is discrete. Remark (technicality) S C is open and closed: Some cocpct latts use algebraic integers not in Z. 1 2 Acc pt p γC γC S C ≠ ∈ ⇒ ∩ 1 2 ∅ Let α √2 (or other algebraic int) and F Q[α]. 1 ≠ = = C γ− s1 snC (no acc pts) ⇒ ∩1 ··· ∅ If G is defined over F and GZ[α] is discrete γ− s1 s S γ S ⇒ ··· n ∈ ⇒ ∈ 1 2 then GZ[α] is an arithmetic subgroup of G. p S C. ⇒ ∈ 1 2 It is a lattice in G. (“Restriction of scalars”) So S C G. Preceding argument: GZ S . 1 2 = = 1 2 Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 9 / 12 Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 10 / 12

Lemma (Selberg) Lemma. The centre of is finite. GZ is virtually torsion-free : finite-index subgrp + GZ, + is torsion-free. Exercise Γ ∃ ⊆ (no nontrivial elements of finite order) Prove when is a cocompact lattice in G SL(n, R). Γ Γ = Proof. Hint. More precisely,Γ show Z( ) Z(G). Natural homo ρ3 : SL(n, Z/3Z), + ker(ρ3). ⊆ → = For , the conjugacy class G is compact. Since / + ρ3( ) SL(n, Z/3Z) is finite, z Z( ) z - ⊆ ∈ Γ it suffices to showΓ + is torsion-free. Γ If a diag(a1,...,a ) is any diagonal(izable) = Γ n Γ Γ Γ k element of , then Let γ +, write γ I 3 T , T 0 (mod 3). G n n n m ∈ k m=Γ + 4≡ (a− za ) z (a /a ) . γ (I 3 T) ij = ij · j i = + m This is bounded, so it must be constant. IΓ m(3kT) 32kT 2 = + + 2 +··· Therefore centralizes . k k $ 1 $ a z I 3 mT (mod( 3) + + ) if 3 m ≡ + k $ 1 $ 1 | is generated by its diagonalizable elements. I (mod 3 + + ) if 3 + % m. G 4≡ Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 11 / 12 Dave Witte Morris (U of Lethbridge) Introduction to Arithmetic Groups Auckland, Feb 2020 12 / 12