aa r X i v : . [ m a t h . G T ] D ec Arithmeticity of complex hyperbolic triangle groups
Matthew Stover ∗ University of Michigan [email protected]
June 15, 2018
Abstract
Complex hyperbolic triangle groups, originally studied by Mostowin building the first nonarithmetic lattices in PU(2 , ( R ), so triangle groups aregenerically nonarithmetic. We prove similar finiteness theorems forcomplex hyperbolic triangle groups that determine an arithmetic lat-tice in PU(2 , In his seminal 1980 paper, Mostow constructed lattices in PU(2 ,
1) gener-ated by three complex reflections [10]. He not only gave a new geometricmethod for building lattices acting on the complex hyperbolic plane, butgave the first examples of nonarithmetic lattices in PU(2 , complex hyperbolic triangle groups . We introduce thesegroups in §
2. See also [5, 17], which, along with [10], inspired much of therecent surge of activity surrounding these groups.Around the same time, Takeuchi classified the Fuchsian triangle groupsthat determine arithmetic lattices in PSL ( R ) [18]. In particular, he provedthat there are finitely many and gave a complete list. Since there are in-finitely many triangle groups acting on the hyperbolic plane discretely with ∗ partially supported by NSF RTG grant DMS 0602191 , angular invariant ψ ∈ [0 , π ). See §
2. In particular,there is a 1-dimensional deformation space of complex triangles with fixedtriple of angles. The typical assumption is that ψ is a rational multiple of π , in which case the angular invariant is called rational . We call it irrational otherwise.When a complex hyperbolic triangle group is also an arithmetic lattice,we will call it an arithmetic complex hyperbolic triangle group. Note thatthis immediately implies discreteness. Our first result is for nonuniformarithmetic complex hyperbolic triangle groups. We prove the following in § Theorem 1.1.
There are finitely many nonuniform arithmetic complex hy-perbolic triangle groups with rational angular invariant. If Γ is a nonuniformarithmetic complex hyperbolic triangle group with irrational angular invari-ant ψ , then e iψ is contained in a biquadratic extension of Q . We emphasize that complex reflection groups are allowed to have gen-erators of arbitrary finite order. A usual assumption is that all generatorshave the same order, a restriction that we avoid. See Theorem 6.1 for a moreprecise formulation of Theorem 1.1. Proving that a candidate is indeed alattice is remarkably difficult, as evidenced in [10, 3], so we do not give adefinitive list. One consequence of the proof (see Theorem 1.5(1) below) isthe following.
Corollary 1.2.
Suppose that Γ is a nonuniform lattice in U(2 , . If Γ contains a complex reflection of order or at least , then Γ is nonarithmetic. In the cocompact setting, the arithmetic is much more complicated.Arithmetic subgroups of U(2 ,
1) come in two types, defined in §
3, oftencalled first and second. In § Theorem 1.3.
Let Γ < U(2 , be a lattice containing a complex reflection.Then Γ contains a Fuchsian subgroup stabilizing the wall of the reflection in H C . We also give a generalization to higher-dimensional complex reflectiongroups. Theorem 1.3 leads to the following, which we also prove in § heorem 1.4. Let Γ < U(2 , be a lattice, and suppose that Γ is commen-surable with a lattice Λ containing a complex reflection. Then Γ is eitherarithmetic of first type or nonarithmetic. In particular, when considering a complex reflection group as a candidatefor a nonarithmetic lattice, one must only show that it is not of the first type.Fortunately, this is the case where the arithmetic is simplest to understand.The effect of the angular invariant is a particular sticking point in gen-eralizing Takeuchi’s methods. In §
5, the technical heart of the paper, westudy the interdependence between the geometric invariants of the triangleand the arithmetic of the lattice. We collect the most useful of these factsas the following. See §§ Theorem 1.5.
Suppose that Γ is an arithmetic complex hyperbolic trianglegroup. Suppose that for j = 1 , , the generators have reflection factors η j ,the complex angles of the triangle are θ j , and that the angular invariant is ψ . Let E be the totally imaginary quadratic extension of the totally real field F that defines Γ as an arithmetic lattice. Then:1. η j ∈ E for all j ;2. cos θ j ∈ F for all j ;3. e iψ ∈ E and cos ψ ∈ F ;4. If θ j ≤ π/ for all j , then cos ψ ∈ Q (cid:0) cos θ , cos θ , cos θ , cos θ cos θ cos θ (cid:1) ; E ⊆ Q (cid:0) cos θ , cos θ , cos θ , e iψ cos θ cos θ cos θ (cid:1) ;6. If ψ is rational, then E is a subfield of a cyclotomic field. In §
6, we use the results from § Theorem 1.6.
Suppose that Γ is an arithmetic complex hyperbolic trianglegroup for which the associated complex triangle is a right triangle. Thenthe angles of the triangle are the angles of an arithmetic Fuchsian trianglegroup. There are finitely many such Γ with rational angular invariant. §
6. This is thecase which has received the most attention, in particular from Mostow [10]and, in the ideal case, by Goldman–Parker [5] and Schwartz [17]. See also[2]. Here we cannot explicitly bound orders of generators, angles, or angularinvariants because our proof relies on asymptotic number theory for whichwe do not know precise constants. Nevertheless, we obtain finiteness in thesituation that has received the greatest amount of attention since Mostow’soriginal paper. See [11, 12, 13, 3] and references therein for more recentexamples of lattices and restrictions on discreteness.
Theorem 1.7.
There are finitely many arithmetic complex hyperbolic equi-lateral triangle groups with rational angular invariant.
We assume some basic knowledge of complex hyperbolic geometry, e.g., thefirst three chapters of [4]. Let V be a three-dimensional complex vectorspace, equipped with a hermitian form h of signature (2 , H C is the space of h -negative lines in V . The metric on H C is defined via h as in [4, Chapter 3], and the action of U(2 ,
1) on H C by isometries descends from its action on V and factors through projectiononto PU(2 , ∂ H C is the space of h -isotropic lines, andwe set H C = H C ∪ ∂ H C .A complex reflection is a diagonalizable linear map R : V → V withone eigenvalue of multiplicity 2 (or, more generally, multiplicity n − V ) = n ). We assume that R has finite order, so the third eigenvalue is aroot of unity η . We call η the reflection factor of R . Decompose V = V ⊕ V η into the 1- and η -eigenspaces, and choose v η ∈ V so that V η = Span C { v η } .We begin with an elementary lemma that will be of use later, keeping inmind that every complex reflection has 1 as an eigenvalue. Lemma 2.1.
Let A ∈ GL n ( C ) be a diagonalizable linear transformation.Let E ⊆ C be a subfield, and suppose that E n has a basis consisting ofeigenvectors for A . Furthermore, suppose that A has at least one eigen-value in E and that there exists x ∈ C × so that xA ∈ GL n ( E ) . Then alleigenvalues of A are in E .Proof. Let v , . . . , v n ∈ E n be a basis of eigenvectors for A , and let λ j bethe eigenvalue associated with v j , 1 ≤ j ≤ n . Without loss of generality, λ ∈ E . Then xA also has eigenvectors v , . . . , v n , and xAv j = xλ j v j ∈ E n j , since xA ∈ GL n ( E ). Then xλ j ∈ E , 1 ≤ j ≤ n . Since λ ∈ E , itfollows that x ∈ E , which implies that λ j ∈ E for all j .Assume that R ∈ U(2 , R acting on H C isthe subset of h -negative lines in V . This is a totally geodesic holomorphicembedding of the hyperbolic plane if and only if V η is an h -positive line.These subspaces are called complex hyperbolic lines . Following [4, § v η a polar vector for R .When V η is h -negative, the fixed set of R on H C is a point, and R issometimes called a reflection through that point. The complex reflectionsin this paper will always be of through complex hyperbolic lines. That is,the η -eigenspace will always be an h -positive line.Let W be the complex hyperbolic line in H C fixed by R . We call thisthe wall of R . If v η is a polar vector, then R is the linear transformation z z + ( η − h ( z, v η ) h ( v η , v η ) v η . (1)We refrain from normalizing the polar vector to have h -norm one, since wewill often choose a polar vector with coordinates in a subfield E of C , and E ⊂ V might not contain an h -norm one representative for the given lineof polar vectors.Now, consider three complex reflections R , R , R ∈ U(2 ,
1) with re-spective distinct walls W , W , W in H C . If v j is a polar vector for R j , then W j and W j +1 (with cyclic indices) meet in H C if and only if h ( W j , W j +1 ) = | h ( v j , v j +1 ) | h ( v j , v j ) h ( v j +1 , v j +1 ) < , (2)The two walls meet at a point z j stabilized by the subgroup of U(2 , R j and R j +1 . The complex angle θ j between W j and W j +1 ,the minimum angle between the two walls, satisfies cos θ j = h ( W j , W j +1 ).The walls W j and W j +1 meet at a point p j in ∂ H C if and only if | h ( v j , v j +1 ) | h ( v j , v j ) h ( v j +1 , v j +1 ) = 1 , (3)so we say that the complex angle is zero. The group generated by R j and R j +1 fixes p j , so it is contained in a parabolic subgroup of U(2 , § { R j } be reflections through walls { W j } , j = 1 , ,
3. When the pair-wise intersections of the walls are nontrivial in H C , they determine a complex riangle in H C , possibly with ideal vertices. The subgroup △ ( R , R , R ) ofU(2 ,
1) generated by the R j s is called a complex hyperbolic triangle group .A complex hyperbolic triangle group is sometimes defined as one withorder two generators, and groups with higher order generators are called generalized triangle groups. We avoid this distinction and do not make theusual assumption that all generators have the same order.Unlike Fuchsian triangle groups, the complex angles { θ , θ , θ } do notsuffice to determine △ ( R , R , R ) up to Isom( H C )-equivalence. We alsoneed to consider Cartan’s angular invariant ψ = arg (cid:0) h ( v , v ) h ( v , v ) h ( v , v ) (cid:1) . (4)A complex triangle is uniquely determined up to complex hyperbolic isome-try by the complex angles between the walls, and the angular invariant. See[1] and [15, Proposition 1]. Up to the action of complex conjugation on H C ,we can assume ψ ∈ [0 , π ].We call the angular invariant rational if ψ = sπ/t for some (relativelyprime) s, t ∈ Z . In other words, the angular invariant is rational if and onlyif e iψ is a root of unity.Let △ ( R , R , R ) be a complex hyperbolic triangle group in U(2 , η j , complex angles θ j , polar vectors v j , j = 1 , , ψ . Suppose that { v , v , v } is a basis for V . Then △ ( R , R , R ) preserves the hermitian form h △ ( R ,R ,R ) = e iψ cos θ e iψ cos θ e − iψ cos θ e iψ cos θ e − iψ cos θ e − iψ cos θ . (5)We denote this by h △ when the generators are clear. U(2 , Let F be a totally real number field, E a totally imaginary quadratic exten-sion, and D a central simple E -algebra of degree d . Let τ : D → D be aninvolution, that is, an antiautomorphism of order two. Then τ is of secondkind if τ | E is the Galois involution of E/F . There are two cases of interest.1. If D = E (i.e., d = 1), then τ is the Galois involution.2. If d = 3, then D is a cubic division algebra with center E .6ee [9] for more on algebras with involution.For d as above, let r = 3 /d . A form h : D r → D is called hermitian or τ - hermitian if it satisfies the usual definition of a hermitian form with τ inplace of complex conjugation. If d = 1, then h is a hermitian form on E asusual. If d = 3, then there exists an element x ∈ D ∗ such that τ ( x ) = x and h ( y , y ) = τ ( y ) xy for all y , y ∈ D .This determines an algebraic group G , the group of elements in GL r ( D )preserving h . For every embedding ι : F → R , we obtain an embedding of G into the real Lie group U( ι ( h )). Let G be the associated projective unitarygroup.If O is a order in D r , then the subgroup Γ O of GL r ( O ) preserving h embeds as a discrete subgroup of G ( R ) = Y ι : F → R U( ι ( h )) . If Γ O is the image of Γ O in G , then Γ O is a discrete subgroup of the associatedproduct of projective unitary groups.The projection of Γ O onto any factor of G ( R ) is discrete if and only if thekernel of the projection of G ( R ) onto that factor is compact. Therefore, weobtain a discrete subgroup of U(2 ,
1) if and only if U( ι ( h )) is noncompactfor exactly one real embedding of F .Then Γ O is a lattice in PU(2 ,
1) by the well-known theorem of Boreland Harish-Chandra. An arithmetic lattice in PU(2 ,
1) is any lattice Γ < PU(2 ,
1) which is commensurable with Γ O for some G as above and an order O in D .Since arithmeticity only requires commensurability with Γ O , studyingan arbitrary Γ in the commensurability class of Γ O requires great care. Theimage of any element γ ∈ Γ in PU(2 ,
1) does, however, have a representativein GL ( E ), that is, if there exists x ∈ C × so xγ ∈ GL ( E ). This followsfrom the fact, due to Vinberg [19], that Γ is F -defined over the adjoint form G , i.e., Q (cid:0) Tr Ad Γ (cid:1) = F. This important fact also follows from [14, Proposition 4.2].
We require some elementary results from the theory of discrete subgroupsof Lie groups. The primary reference is [16]. Let G be a second countable,locally compact group and Γ < G a lattice. Recall that G/ Γ carries a7nite G -invariant measure and Γ is uniform in G if G/ Γ is compact. For asubgroup
H < G , we let Z G ( H ) denote the centralizer of H in G . We needthe following two results. Lemma 4.1 ([16] Lemma 1.14) . Let G be a second countable locally compactgroup, Γ < G a lattice, ∆ ⊂ Γ a finite subset, and Z G (∆) the centralizer of ∆ in G . Then, Z G (∆)Γ is closed in G . Theorem 4.2 ([16] Theorem 1.13) . Let G be a second countable locallycompact group, Γ < G be a uniform lattice, and H < G be a closed subgroup.Then H Γ is closed in G if and only if H ∩ Γ is a lattice in H . We now use the above to prove Theorem 1.3.
Proof of Theorem 1.3.
Assume that Γ is a cocompact arithmetic lattice inU(2 ,
1) containing a complex reflection and that ∆ is the subgroup of Γgenerated by this reflection. The centralizer of ∆ in U(2 ,
1) is isomorphicto the extension of U(1 ,
1) by the center of U(2 , ,
1) of the wall of the reflection that generates ∆. It follows from Lemma4.1 and Theorem 4.2 that Γ ∩ U(1 ,
1) is a lattice. Since any sublattice ofan arithmetic lattice is arithmetic, Γ contains a totally geodesic arithmeticFuchsian subgroup.
Proof of Theorem 1.4.
A totally geodesic arithmetic Fuchsian group comesfrom a subalgebra of D r , with notation as in §
3. When Γ is of second type, D is a cubic division algebra. The totally geodesic Fuchsian group wouldcorrespond to a quaternion subalgebra of D , which is impossible. When Γ isof first type, this quaternion subalgebra corresponds to rank 2 subspaces of E on which h has signature (1, 1). Therefore, Γ contains complex reflectionsif and only if Γ is of first type. Remark.
One can also prove Theorem 1.4 using the structure of unit groupsof division algebras.We now briefly describe how these results generalize to reflections actingon higher-dimensional complex hyperbolic spaces. If Γ < U( n,
1) is a lattice,an element R ∈ Γ is a codimension s reflection if it stabilizes a totallygeodesic embedded H n − s C and acts by an element of the unitary group of thenormal bundle to the wall. If Γ is arithmetic, the associated algebraic groupis constructed via a hermitian form on D r , where D is a division algebra ofdegree d with involution of the second kind over a totally imaginary field E ,and where rd = n + 1. 8 heorem 4.3. Suppose Γ < U( n, is a cocompact arithmetic lattice withassociated algebraic group coming from a hermitian form on D r , where D isa central simple algebra with involution of the second kind. If Γ contains acodimension s reflection, then Γ contains a cocompact lattice in U( n − s, .Also, n − s + 1 = ℓd for some < ℓ ≤ r and the associated algebraic subgroupcomes from a hermitian form on D ℓ . Corollary 4.4.
Let Γ < U( n, be an arithmetic lattice generated by com-plex reflections through totally geodesic walls isometric to H n − C . Then Γ is of so-called first type, i.e., the associated algebraic group is the auto-morphism group of a hermitian form on E n +1 , where E is some totallyimaginary quadratic extension of a totally real field. In this section, we relate the geometric invariants of a complex triangle tothe arithmetic invariants of the complex reflection group. It is the technicalheart of the paper.Let Γ = △ ( R , R , R ) be a complex hyperbolic triangle group withreflection factors η j , complex angles θ j , and angular invariant ψ . Assumethat Γ is an arithmetic lattice in U(2 , h over a totally imaginary field E .Let F be the totally real quadratic subfield of E . Lemma 5.1.
We can choose polar vectors v j for the reflection R j so that v j ∈ E .Proof. Associated with each reflection is an arithmetic Fuchsian subgroupof Γ. When Γ is a uniform lattice, this follows from Theorem 1.3. For thenonuniform case, see [6, Chapter 5]. Arithmetic Fuchsian subgroups stabi-lizing a complex hyperbolic line come from the h -orthogonal complement ofan h -positive line in E . (To be more precise, this line is h -positive overthe unique real embedding of F at which h is indefinite.) Any vector in E representing this line is a polar vector for R j .This leads us to the following important fact. Lemma 5.2.
Each reflection factor η j is contained in E . roof. It follows from Proposition 4.2 in [14] that there exists an x j ∈ C sothat x j R j ∈ GL ( E ) (see the end of § h -orthogonal complement to a polar vector evidently has an E -basis, E has a basis of eigenvectors for R j . The lemma follows from Lemma 2.1.Now we turn to the complex angles and the angular invariant. Lemma 5.3.
For each j , cos θ j ∈ F and e iψ ∈ E .Proof. Choose polar vectors v j ∈ E . The terms in Equations (2) and (3)resulting from these choices of polar vectors are all contained in E . Hencecos θ j ∈ F . One can also prove this using Tr Ad( R R ) and Lemma 5.2.Similarly, consider δ = h ( v , v ) h ( v , v ) h ( v , v ) = re iψ ∈ E from Equation (4). Note that e iψ ∈ E if and only if r ∈ E . Either way,when δ = 0, we have δ/δ = e iψ ∈ E . This completes the proof.Combining the above, we see that Q (cid:0) η , η , η , cos θ , cos θ , cos θ , e iψ (cid:1) ⊆ E. We can also bound E from above using the fact that E ⊆ Q (cid:0) Tr Γ (cid:1) . Usingwell-known computations of traces for products of reflections (e.g., [10, § Q (cid:0) Tr Γ (cid:1) = Q ( η , η , η , cos θ , cos θ , cos θ , e iψ cos θ cos θ cos θ (cid:1) . Similarly, Q (cid:0) Re( η ) , Re( η ) , Re( η ) , cos θ , cos θ , cos θ , cos ψ (cid:1) ⊆ F ⊆ Q (Re( η ) , Re( η ) , Re( η ) , cos θ , cos θ , cos θ , cos ψ cos θ cos θ cos θ (cid:1) . This gives the following.
Corollary 5.4.
Let Γ be a complex hyperbolic triangle group and an arith-metic lattice in U(2 , . If the angular invariant of the triangle associatedwith Γ is rational, then the fields that define Γ as an arithmetic lattice aresubfields of a cyclotomic field. h △ be as in (5) and consider h △ as a hermitian form on the extension E △ = Q (cid:0) η , η , η , cos θ , cos θ , cos θ , e iψ (cid:1) , of E . It follows from [10, §
2] that h and h △ are equivalent over E △ . Conse-quently, h △ is indefinite over exactly one complex conjugate pair of placesof E . This implies that there are precisely [ E △ : E ] conjugate pairs of placesof E △ over which h △ is indefinite.Let H be a hermitian in 3 variables over the complex numbers for whichthere is a vector with positive H -norm. Then H is indefinite if and only ifdet( H ) <
0. Since any polar vector has positive h △ -norm by definition, wehave the following. Proposition 5.5.
There are exactly [ E △ : E ] complex conjugate pairs ofGalois automorphisms τ of E △ ⊂ C under which τ (det( h △ )) is negative.All such automorphisms act trivially on E . This has the following consequence for the relationship between the ge-ometry of the triangle and the arithmetic of the lattice.
Corollary 5.6. If Γ is a complex hyperbolic triangle group and an arithmeticlattice, then the reflection factors of Γ are restricted by the geometry of thetriangle. In particular, E △ = Q (cid:0) cos θ , cos θ , cos θ , e iψ (cid:1) . Proof.
Since det( h △ ) is independent of the reflection factors, for each Galoisautomorphism of E △ / Q (cid:0) cos θ , cos θ , cos θ , e iψ (cid:1) we obtain a new complex conjugate pair of embeddings of E △ into C suchthat det( h △ ) is negative. Any such automorphism necessarily acts nontriv-ially on some reflection factor η j . These embeddings of E △ lie over differentplaces of E by Lemma 5.2. This contradicts Proposition 5.5.We also obtain the following dependence between the angular invariantand the angles of the triangle. Proposition 5.7. If Γ is a complex hyperbolic triangle group and an arith-metic lattice. If Γ has rational angular invariant and θ j ≤ π/ for j = 1 , , ,then cos ψ ∈ F ′ = Q (cid:0) cos θ , cos θ , cos θ , cos θ cos θ cos θ (cid:1) . roof. If ψ is rational, then E △ is a subfield of a cyclotomic field K N = Q (cid:0) ζ N (cid:1) , where ζ N is a primitive N th root of unity. Therefore the Galoisautomorphisms of E △ are induced by ζ N ζ mN for some m relatively primeto N .Consider the stabilizer S of F ′ in Gal( K N / Q ). It acts on the roots ofunity in E △ as a group of rotations along with complex conjugation. Bydefinition of E △ , every nontrivial element of S acts nontrivially on e iψ . Inparticular, if cos ψ / ∈ Q and S contains a rotation through an angle otherthan an integral multiple of π , then the orbit of e iψ under S contains twonon-complex conjugate points with distinct negative real parts.Let τ be any such automorphism of E △ . Then, since τ (cos θ j ) = cos θ j for all j by definition of S , τ (det( h △ )) = 1 − X j =1 cos θ j + 2 τ (cos ψ ) Y j =1 cos θ j . Furthermore, 1 − P cos θ j ≤ θ j = π/r j that arethe angles of a hyperbolic triangle with each r j ≥
3. Since τ (cos ψ ) < θ j >
0, it follows that τ (det h △ ) <
0. Since τ acts nontrivially on e iψ ∈ E , this contradicts Proposition 5.5. Therefore, S is generated bycomplex conjugation and rotation by π , so cos ψ ∈ F ′ . Remark.
For several of Mostow’s lattices in [10], F ′ = F (with notationas above) and cos ψ / ∈ F ′ . Thus Proposition 5.7 is the strongest possibleconstraint on rational angular invariants. We are now prepared to collect facts from § Theorem 6.1.
Suppose that Γ is a complex hyperbolic triangle group and anonuniform arithmetic lattice in U(2 , . Then1. Each generator has order , , , or .2. Each complex angle θ j of the triangle comes from the set { π/ , π/ , π/ , π/ , } .
3. If ψ is the angular invariant, then e iψ lies in a biquadratic extensionof Q . . If ψ is rational, then ψ = sπ/t for t ∈ { , , , , , } . Proof.
Since Γ is a nonuniform arithmetic lattice, the associated field E isimaginary quadratic. For , we apply Lemma 5.2 to E . For and , weapply Lemma 5.3. Then follows from determining those integers m sothat ϕ ( m ) = 2 or 4 and e iψ is at most quadratic over Q , where ϕ is Euler’stotient function.See [13, 3] for the known nonuniform arithmetic complex hyperbolictriangle groups. We now determine the right triangle groups that can de-termine an arithmetic lattice in SU(2 , Proof of Theorem 1.6.
Suppose that Γ is an arithmetic complex hyperbolictriangle group with θ = π/
2. The hermitian form h △ associated with thetriangle has determinant 1 − cos θ − cos θ . By Lemma 5.3, this is an element of the totally real field F that defines Γas an arithmetic lattice. Consequently, there is no Galois automorphism of F over Q under which this expression remains negative.This is precisely Takeuchi’s condition that determines whether or notthe triangle in the hyperbolic plane with angles π/ , θ , θ determines anarithmetic Fuchsian group. The theorem follows from Takeuchi’s classifica-tion of arithmetic Fuchsian right triangle groups, Lemma 5.3, and Corollary5.6 There are 41 such right triangles in H . We now finish the paper withfiniteness for arithmetic complex hyperbolic triangle groups with equilateralcomplex triangle and rational angular invariant. Proof of Theorem 1.7.
Let Γ be an arithmetic complex hyperbolic trianglegroup with equilateral triangle of angles π/n and angular invariant ψ . ByProposition 5.7, we can assume that ψ = sπ/ n for some integer s . In-deed, F ′ = Q (cos π/n ), and the assertion follows from an easy Galois theorycomputation.Then det( h △ ) = 1 − ( π/n ) + 2 cos( sπ/ n ) cos ( π/n ) , (6)13o we want to find a nontrivial Galois automorphism of F △ whose restrictionto F is nontrivial and such that the image of (6) under this automorphismis negative. Let p be the smallest rational prime not dividing 12 n . Thisdetermines a nontrivial Galois automorphism τ p of F △ under which τ p (det( h △ )) = 1 − ( pπ/n ) + 2 cos( psπ/ n ) cos ( pπ/n ) . (7)It is nontrivial on F by definition. If we show that τ p (det( h △ )) < n sufficiently large, this, along with Corollary 5.6, suffices to prove the theo-rem.First, notice that the function D ( x, y ) = 1 − +2 cos y cos x is an increasing function of x ∈ (0 , π/
2) for any fixed y . In our language, thisimplies that if y is the angular invariant of an equilateral complex trianglein H C with angle x , then it remains an angular invariant for a complextriangle with angle x ′ for any x ′ < x . Similarly, if we know that π/ n is anangular invariant for a triangle with angles pπ/n , then we know that psπ/n (more precisely, a representative modulo 2 π ) is the angular invariant of anequilateral triangle in H C with angles pπ/n . Therefore, it is enough to showthat π/ n is the angular invariant of a triangle having angles pπ/n for allsufficiently large n , where p is the smallest not prime dividing 12 n .From the above, we conclude further that it suffices to show that thereexists a function q ( n ) such that p < q ( n ) and1 − ( q ( n ) π/n ) + 2 cos( π/ n ) cos ( q ( n ) π/n ) < n . To prove this, we consider the function j ( n ),defined by Jacobsthal [8]. For any integer n , j ( n ) is the smallest integersuch that any j ( n ) consecutive integers must contain one that is relativelyprime to n . Clearly p ≤ j (12 n ).Iwaniec [7] proved that j ( n ) ≪ (log n ) . Therefore, for any ǫ >
0, there is an n ǫ so that the first prime numbercoprime to 12 n is at most log(12 n ) ǫ for every n ≥ n ǫ . Consider thefunction f ǫ ( x ) = 1 − (cid:0) log(12 /x ) ǫ πx (cid:1) + 2 cos (cid:0) πx/ (cid:1) cos (cid:0) log(12 /x ) ǫ πx (cid:1) . x → f ǫ ( x ) exists and equals 0 for all ǫ >
0. Further, x = 0 is a localmaximum of f ǫ , so f ǫ (1 /n ) < n .Taking q ( n ) = log( n ) ǫ for any small ǫ shows that (8) holds for allsufficiently large n . This implies that (7) is negative for all large n . Thisproves the theorem.Unfortunately, the proof of Theorem 1.7 isn’t effective, so we cannot listthe angles that can possibly determine an arithmetic lattice. In particular,we don’t know which n makes the bound from [7] effective for any ǫ > n = 10 for some ǫ , which computer experimentsshow is extraordinarily likely, then we obtain n < , , H that defines an arithmetic Fuchsian group hasangles π/ Acknowlegments
I thank the referee for several helpful comments.
References [1] Ulrich Brehm. The shape invariant of triangles and trigonometry intwo-point homogeneous spaces.
Geom. Dedicata , 33(1):59–76, 1990.[2] Martin Deraux. Deforming the R -Fuchsian (4 , , Topology , 45(6):989–1020, 2006.[3] Martin Deraux, John Parker, and Julien Paupert. Census of the com-plex hyperbolic sporadic groups. To appear in Experiment. Math.[4] William M. Goldman.
Complex hyperbolic geometry . Oxford UniversityPress, 1999.[5] William M. Goldman and John R. Parker. Complex hyperbolic idealtriangle groups.
J. Reine Angew. Math. , 425:71–86, 1992.[6] Rolf-Peter Holzapfel.
Ball and surface arithmetics . Friedr. Vieweg &Sohn, 1998.[7] Henryk Iwaniec. On the problem of Jacobsthal.
Demonstratio Math. ,11(1):225–231, 1978. 158] Ernst Jacobsthal. ¨Uber Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist. Norke Vid. Selsk. Forh. Trondheim , 33:117–124, 1961.[9] Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-PierreTignol.
The book of involutions . American Mathematical Society, 1998.[10] G. D. Mostow. On a remarkable class of polyhedra in complex hyper-bolic space.
Pacific J. Math. , 86(1):171–276, 1980.[11] John R. Parker. Unfaithful complex hyperbolic triangle groups. I. In-volutions.
Pacific J. Math. , 238(1):145–169, 2008.[12] John R. Parker and Julien Paupert. Unfaithful complex hyperbolic tri-angle groups. II. Higher order reflections.
Pacific J. Math. , 239(2):357–389, 2009.[13] Julien Paupert. Unfaithful complex hyperbolic triangle groups. III.Arithmeticity and commensurability.
Pacific J. Math. , 245(2):359–372,2010.[14] Vladimir Platonov and Andrei Rapinchuk.
Algebraic groups and numbertheory . Academic Press Inc., 1994.[15] Anna Pratoussevitch. Traces in complex hyperbolic triangle groups.
Geom. Dedicata , 111:159–185, 2005.[16] M. S. Raghunathan.
Discrete subgroups of Lie groups . Springer-Verlag,1972.[17] Richard Evan Schwartz. Complex hyperbolic triangle groups. In
Pro-ceedings of the International Congress of Mathematicians, Vol. II (Bei-jing, 2002) , pages 339–349. Higher Ed. Press, 2002.[18] Kisao Takeuchi. Arithmetic triangle groups.
J. Math. Soc. Japan ,29(1):91–106, 1977.[19] `E. B. Vinberg. Rings of definition of dense subgroups of semisimplelinear groups.