Universal systole bounds for arithmetic locally symmetric spaces
aa r X i v : . [ m a t h . DG ] F e b UNIVERSAL SYSTOLE BOUNDSFOR ARITHMETIC LOCALLY SYMMETRIC SPACES
SARA LAPAN, BENJAMIN LINOWITZ, AND JEFFREY S. MEYER
Abstract.
The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes ofsimple arithmetic locally symmetric orbifolds. We establish new bounds for the translationlength of semisimple x ∈ SL n ( R ) in terms of its associated Mahler measure. We use thesegeometric methods to prove the existence of extensions of number fields in which fixed setsof primes have certain prescribed splitting behavior. Introduction
The systole of a closed Riemannian manifold is the minimal length of a non-contractibleclosed loop. In this paper, we investigate lower bounds for the systoles of simple arithmeticlocally symmetric orbifolds.To give some history and motivation, we begin by describing the situation in dimension2. If Γ ⊂ SL ( R ) is a lattice and H is the (real) hyperbolic plane, then S := H / Γ is ahyperbolic surface (possibly with cusps or cone points). An element x ∈ SL ( R ) is hyperbolic if it is diagonalizable with positive real eigenvalues. Let { a, a − } (where a >
1) denote theeigenvalues of x . If x ∈ Γ, then it corresponds to a closed geodesic in S with length(1.1) ℓ ( x ) := 2 log | a | . It is well known ([14, § § a is an algebraic integer greater than 1 which is Galois conjugateto its inverse and for which all other Galois conjugates have norm 1, or in other words, a is a Salem number . Conversely, Chinburg, Neumann, and Reid showed that every Salem numberappears this way for some arithmetic hyperbolic surface.The
Mahler measure of a monic polynomial p ( X ) = Q ni =1 ( X − a i ) ∈ C [ X ] is the product M ( p ) := Q ni =1 max { , | a i |} . The Mahler measure is a fascinating function on polynomialsthat appears throughout algebra, geometry, and dynamics (see for example the surveys [6]and [20]). If p x ( X ) is the characteristic polynomial of x , then it follows that (1.1) can berewritten in terms of the Mahler measure of p x ,(1.2) ℓ ( x ) = 2 log( M ( p x )) . Equation (1.2) provides a deep connection between length and the Mahler measure. If p isa cyclotomic polynomial, then M ( p ) = 1. Lehmer’s Problem asks: is the Mahler measure ofirreducible non-cyclotomic monic polynomials with integer coefficients bounded away from1? The polynomial(1.3) L ( X ) = X + X − X − X − X − X − X + X + 1found by Lehmer (and now called Lehmer’s polynomial) has Mahler measure M ( L ) ≈ . M ( L ) is the smallest known nontrivial Mahler measure and it is stillan open question if in fact this is the smallest over all monic polynomials in Z [ X ]. It follows S. Lapan, B. Linowitz and J. Meyer that Lehmer’s problem implies the open
Short Geodesic Problem , which asks: is there auniversal lower bound for the systoles of arithmetic hyperbolic surfaces?In [5], Emery, Ratcliffe, and Tschantz extended the identification (1.2) to n -dimensionalarithmetic hyperbolic orbifolds of simplest type. In particular, they were able to get systolebounds in terms of Salem numbers of bounded degree.Our goal in this paper is to generalize these results and establish lower bounds for broadclasses of arithmetic locally symmetric spaces. In order to achieve this goal we need a gener-alization of (1.2) to semisimple x ∈ SL n ( R ). In Section 2, we establish such a generalizationwith our Mahler-Length Bounds Theorem A below. Theorem A (Mahler-Length Bounds) . If n ≥ , x ∈ SL n ( R ) is semisimple, and p x denotesthe characteristic polynomial of x , then (1.4) 2 r n log( M ( p x )) ≤ ℓ ( x ) ≤ M ( p x )) . Let { a , . . . , a n } denote the eigenvalues of x . The lower bound is an equality exactly whenthere exists an r ∈ R such that for all i , either | a i | = r or | a i | − = r , and the upper boundis an equality when | a i | = 1 for all but at most two of the i . Remark 1.1.
Let x be such that in (1.4) the upper bound is an equality and at least oneof its eigenvalues has modulus not equal to 1. Since p x has real coefficients, if a i is complex,its complex conjugate a i is also an eigenvalue. By our assumption on x , it has preciselytwo eigenvalues of modulus not equal to 1, and as the eigenvalues must multiply to 1, itfollows that x or − x has a unique real eigenvalue greater than 1, say α . When p x has integercoefficients, then α is a Salem number. Remark 1.2.
Our Mahler-Length Bounds Theorem A is a generalization of (1.2) to de-gree n ≥ ℓ ( x ) = 2 arccosh( tr x ) to degree n ≥ Theorem B (Universal Systole Bounds) . If N is a simple arithmetic orbifold with field ofdefinition k and universal cover X , then (1.5) sys( N, g ) ≥ √ √ n f ( n ) , where sys( N, g ) denotes the systole of N (as measured relative to the subspace metric (3.1) ), n = | k : Q | dim(Isom( X )) , and f ( n ) = log( M ( L )) > . if n ≤ , (cid:16) log log n log n (cid:17) if n > , (1.6) where L ( X ) = X + X − X − X − X − X − X + X + 1 is Lehmer’s polynomial. To prove this result, we show in Proposition 3.2 that the special linear group is “universal”in the sense that if N ∼ = X / Γ where Γ ⊂ Isom( X ) is arithmetic, then Γ injects into a speciallinear group of explicitly bounded degree and each matrix in the image of this injection hasa characteristic polynomial with integer coefficients. niversal Systole Bounds 3Remark 1.3. Fix a globally symmetric space X of noncompact type. When N ∼ = X / Γ whereΓ is a principal arithmetic lattice in the sense of [18, 3.4], then as the volume of N increases,Prasad’s volume formula [18, 3.11] implies the degree | k : Q | increases, and hence n in (1.5)increases, thereby decreasing the systole bound. Hence Theorem B implies there is a fixeduniversal lower systole bound for orbifolds of bounded volume, but as volume is allowed toincrease, this bound goes to 0.In [10, Theorem B] we showed that every simple arithmetic N has a finite degree cover N ′ such that sys( N ′ ) ≥ c log(Vol( N ′ )) + c for certain constants c and c . In Section 4 weprove that certain covers will always attain systoles with order of magnitude log of volume. Theorem C (Simple Arithmetic Manifolds with Large Systoles) . If N is a compact sim-ple arithmetic manifold, then N has a finite sheeted cover N ′ whose systole has order ofmagnitude log(Vol( N ′ )) . A particularly interesting specialization of Theorem B is to the case of special linear orbifoldsderived from central simple algebras. In Section 5 we give the construction of special linearorbifolds derived from central simple algebras and give an alternate proof of Theorem Dbelow.
Theorem D (Standard Special Linear Short Geodesic Bounds) . If M is a standard speciallinear orbifold of degree n , n ≥ , which is derived from a central simple algebra, then sys( M ) ≥ √ √ n f ( n ) , (1.7) where f ( n ) is as in Theorem B. Interestingly, the results of this paper can be used to prove a result from algebraic numbertheory: for any fixed totally real number field k and finite set S of primes of k , there existsa prime degree field extension of k in which no prime of S splits completely and whose normof relative discriminant may be bounded from above. Although such a result follows fromquantitative refinements of the (deep) Grunwald-Wang theorem (see [23] and [24]), we areable to give a short, geometrically-inspired proof. Theorem E (Existence of Number Fields with Prescribed Arithmetic Properties) . Let p > be a prime number, k be a totally real number field and p , . . . , p s be primes of k . There existsa field extension L/k of degree p in which none of the primes p i split completely and whichhas norm of relative discriminant bounded above by cd c ′ k (cid:0)Q si =1 N k/ Q ( p i ) (cid:1) c ′′ where c, c ′ , c ′′ arepositive constants depending only on p and the degree over Q of k . Acknowledgements.
The authors would like to thank Ralf Spatzier for his useful commentsregarding the proof of Proposition 4.1. The work of the second author is partially supportedby NSF Grant Number DMS-1905437. The third author acknowledges support from U.S.National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GeometricStructures and Representation Varieties” (the GEAR Network).2.
Mahler Measure and Translation Length Inequalities - Theorem A
Throughout this section, we will let p ( X ) = Q ni =1 ( X − a i ) ∈ R [ X ] denote a monic degree n polynomial. We will prove results bounding its translation length (2.6) by its Mahler measure(Theorem 2.3), length, and discriminant (Corollary 2.5). By specializing these results to the S. Lapan, B. Linowitz and J. Meyer case when p ( X ) = p x ( X ) is the characteristic polynomial for some semisimple x ∈ SL n ( R ),we obtain Theorem A.We remind the reader that the Mahler measure of p is the product(2.1) M ( p ) := n Y i =1 max { , | a i |} . For a detailed survey on the Mahler measure, see [20]. While it is open as to whether theMahler measure is uniformly bounded away from 1 for monic, irreducible, noncyclotomicpolynomials in Z [ X ], we do have such a result when degree is fixed. Proposition 2.1.
There exists an explicit monotonically decreasing function on the domainof integers n ≥ , such that f ( n ) > and if p ( X ) ∈ Z [ X ] is a degree n , monic polynomialfor which at least one of its irreducible factors is noncyclotomic, then log( M ( p )) ≥ f ( n ) .Furthermore, f ( n ) can be taken to be (2.2) f ( n ) = log( M ( L )) > . if n ≤ , (cid:16) log log n log n (cid:17) if n > , where L ( X ) = X + X − X − X − X − X − X + X + 1 is Lehmer’s polynomial.Proof. Lower bounds for when p ( X ) is irreducible are well known, such as the bounds ofVoutier [22] f V ( n ) = (cid:16) log log n log n (cid:17) . A check via Sage verifies that f V (15) < . f V ( n ) is monotonically decreasing for all n ≥
15. Meanwhile, Boyd [3, 4] enumerated allinteger polynomials of degree ≤
20 of small Mahler measure and verified that Lehmer’s poly-nomial L attains the minimal Mahler measure in this range. It can be computed in Sage thatlog( M ( L )) > . f ( x ) to be the maximum of the two. When p is not irre-ducible, the proposition follows from the fact that the log of the Mahler measure is additivetogether with the monotonicity of f ( n ). More precisely, if p ( X ) = g ( X ) · · · g s ( X ), wherethe g i are irreducible of degree n i , and without loss of generality, g ( X ) is noncyclotomic,thenlog( M ( p )) = log( M ( g )) + · · · + log( M ( g s )) ≥ f ( n ) + · · · + f ( n s ) ≥ f ( n ) ≥ f ( n ) . (cid:3) We may write p ( X ) as p ( X ) = n X j =0 ( − j s j X n − j (2.3)where s j denotes the j th elementary symmetric polynomial in the roots of p . Let p k := P i a ki .Newton’s identities [8] state that for all 1 ≤ j ≤ n , js j = j X i =1 ( − i − s j − i p i . (2.4)The following result will be used in the proof of Theorem A to bound the Mahler measure. niversal Systole Bounds 5Lemma 2.2. Suppose the polynomial p ( X ) = Q ni =1 ( X − c i ) = P nj =0 ( − j s j X n − j has all realzeros. The coefficient s j is the j th elementary symmetric polynomial in the zeros of p . For k ∈ N , let p k = P ni =1 c ki . Then: (2.5) 2 nn − s ≤ s ≤ np with equality on both sides exactly when c i = c j for all i, j . If all c i ≥ , then p ≤ s withequality exactly when c i = 0 for all i .Proof. By the Cauchy-Schwarz inequality: s = n X i =1 ( c i )(1) ! ≤ n X i =1 c i ! n X i =1 ! = n n X i =1 c i = np , with equality exactly when the sequence { c i } ni =1 and { } ni =1 are linearly dependent. Inparticular, when c i = c j for all i, j . Also, s = n X i =1 c i ! = n X i =1 c i + 2 X i 0, notice that s ≥ p with equality precisely when all c i = 0. Rearranging theprevious equation and combining it with the special case of the Cauchy-Schwarz inequalitygiven above we get: s ≤ n ( s − s ) ⇒ ns ≤ ( n − s . (cid:3) Being motivated by thinking of p as a characteristic polynomial of a semisimple matrix, wedefine the translation length of the polynomial p to be(2.6) ℓ ( p ) := vuut n X i =1 | a i | ) . Theorem 2.3. Let p ( X ) = Q ni =1 ( X − a i ) ∈ R [ X ] and assume Q ni =1 a i = 1 . Then: (2.7) 2 r n log( M ( p )) ≤ ℓ ( p ) ≤ M ( p )) . The lower bound is an equality exactly when there exists an r ∈ R such that for all i , either | a i | = r or | a i | − = r . The upper bound is an equality when | a i | = 1 for all but at most twoof the i . Remark 2.4. When n = 2, the upper and lower bounds in (2.7) are equal and so ℓ ( x ) =2 log( M ( p )), which coincides with (1.2). Proof. Let p − ( X ) := Q ni =1 ( X − a − i ). Since Q ni =1 a i = 1, M ( p ) = M ( p − ) > M ( p ) = p M ( p ) M ( p − ). Then:log( M ( p )) = 12 log( M ( p ) M ( p − )) = 12 n X i =1 (cid:0) log + | a i | + log + (cid:12)(cid:12) a − i (cid:12)(cid:12)(cid:1) = 12 n X i =1 | log | a i || , S. Lapan, B. Linowitz and J. Meyer where log + ( a ) = max { , log( a ) } . Applying Lemma 2.2 to the sequence with { c i } ni =1 = {| log | a i ||} ni =1 , we get:2 log( M ( p )) = s ≤ √ np = vuut n n X i =1 | log | a i || = r n ℓ ( p ) , (2.8)with equality precisely when, for all i , | log | a || = | log | a i || or, equivalently, when | a i | ∈{| a | , | a | − } .We now find an upper bound for ℓ ( p ) by using specific information about the relative sizes ofthe a i that comes from our setup. Assume that | a i | > i , since otherwise | a i | = 1for all i and so ℓ ( p ) = 0 = log( M ( p )). Re-order the { a i } so that | a | ≥ . . . ≥ | a k | ≥ > | a k +1 | ≥ . . . ≥ | a n | > Q ki =1 | a i | = M ( p ) = M ( p − ) = Q ni = k +1 (cid:12)(cid:12) a − i (cid:12)(cid:12) . Using the samenotation as in Lemma 2.2, we want to find a tight upper bound on p in terms of s . Weknow that p = s − s ≤ s since c i := | log | a i || ≥ 0. To get a tighter bound, we find alower bound on s specific to our setup: s = X i The length inequality follows directly from replacing M in Theorem 2.3 with L using(2.12). To get the discriminant inequality, first re-arrange (2.13) and take the logarithm toget: log + (cid:12)(cid:12) n − n disc( p ) (cid:12)(cid:12) ≤ n − 1) log( M ( p )) . Combine that with the lower bound in Theorem 2.3 that 2 log( M ( p )) ≤ p n ℓ ( p ) to get:log + (cid:12)(cid:12) n − n disc( p ) (cid:12)(cid:12) ≤ ( n − r n ℓ ( p ) . (cid:3) Characteristic Polynomials and the Proof of Theorem B If X is a globally symmetric space of noncompact type, then the identity component of itsisometry group G := Isom ( X ) is a connected, adjoint semisimple Lie group and the stabilizersubgroup K = Stab G ( o ) of a point o ∈ X is a maximal compact subgroup [7, IV.3.3, VI.1.1,VI.2.2]. It follows that K\G ∼ = X . We say X is simple if G is simple (i.e., the complexificationof its Lie algebra is simple). By a simple locally symmetric space , we mean a space of theform X / Γ where X is simple and Γ ⊂ G is a lattice.We now discuss the construction of arithmetic lattices in G . Here we assume some familiaritywith algebraic groups and arithmetic groups. For detailed references on algebraic and arith-metic groups, we refer the reader to [1, 15]. For a discussion of arithmetic simple orbifolds,see [10, Section 7].To begin, fix:(1) a degree d , totally real number field k with ring of integers O k ,(2) an embedding k ⊂ R ,(3) a simple, semisimple, adjoint, algebraic k -group G of dimension d such that theidentity component of its real points (G( R )) ∼ = G as Lie groups,(4) a choice of a k -rational basis for its Lie algebra g and use it to identify GL( g ) withGL d ( k ),(5) G( O k ) := Ad − (Ad(G( k )) ∩ GL d ( O k )).Assume G is R -isotropic but G is anisotropic for all non-identity embeddings σ : k → R .Then any Γ ⊂ G( R ) commensurable to G( O k ) is an arithmetic lattice in G( R ).Fix a basis for k over Q and let reg : k → M d ( Q ) be the regular representation associatedto this basis. Let L/k be the Galois closure of k with fixed embedding L ⊂ C . We have twoinjective ring maps ι , ι : M d ( k ) → M d d ( L ) defined below. We obtain our first ring mapby applying the regular representation to each entry of the matrix. ι : M d ( k ) → M d d ( Q ) ⊂ M d d ( L ) S. Lapan, B. Linowitz and J. Meyer y · · · y d . . . y d · · · y d d reg( y ) · · · reg( y d ). . .reg( y d ) · · · reg( y d d ) . Meanwhile, we obtain our second injective ring map by looking at the diagonal map acrossall real embeddings of k , which we denote V ∞ . ι : M d ( k ) → Y σ ∈ V ∞ M d ( L ) ⊂ M d d ( L ) y · · · y d . . . y d · · · y d d σ ( y ) · · · σ ( y d ). . . σ ( y d ) · · · σ ( y d d ) 0 00 . . . 00 0 σ d ( y ) · · · σ d ( y d ). . . σ d ( y d ) · · · σ d ( y d d ) . Lemma 3.1. The images ι (M d ( k )) and ι (M d ( k )) are GL d d ( L ) -conjugate.Proof. If A := L ⊗ k M d ( k ), then A is a central simple L -algebra [19, 7.8]. Letting B :=M d d ( L ), and letting ι and ι extend linearly to A , we have L ⊂ ι ( A ) ⊂ B and L ⊂ ι ( A ) ⊂ B . It then follows from the Skolem–Noether Theorem [19, 7.21] that the images ι ( A ) and ι ( A ) are identified by an inner automorphism of A . More precisely, there exists a g ∈ GL d d ( L ) such that if h ∈ M d ( k ), then ι (1 ⊗ h ) = gι (1 ⊗ h ) g − . The result follows. (cid:3) If H is an algebraic k -group, the restriction of scalars group Res k/ Q (H) is a Q -group suchthat H( k ) ∼ = Res k/ Q (H)( Q ) [17, 2.1.2]. Fix the Q -rational map ϕ : Res k/ Q (SL d ) → SL d d which at the level of k -points is ι , i.e. if y ∈ SL d ( k ) ∼ = Res k/ Q (SL d )( Q ), then ϕ ( y ) = ι ( y ).As in [10, Proposition 7.2], composing the adjoint map with restriction of scalars, we obtainan embedding of Lie groups ρ : G( R ) → SL d d ( R ). This embedding has many nice featuresand in Proposition 3.2 we now establish an important one concerning characteristic poly-nomials: relative to this embedding, semisimple elements of arithmetic lattices always havecharacteristic polynomials with integer coefficients. Proposition 3.2. If Γ ⊂ G( R ) is an arithmetic lattice and x ∈ Γ is semisimple, then p Ad( x ) ( X ) ∈ O k [ X ] and p ρ ( x ) ( X ) ∈ Z [ X ] .Proof. To begin, we show that if x ∈ Γ is semisimple, then the characteristic polynomial p Ad( x ) ( X ) of the matrix Ad( x ) ∈ SL d ( k ) lies in O k [ X ]. By assumption, Γ is commensurablewith G( O k ), so let e = [Γ : (Γ ∩ G( O k ))]. Then x e ∈ G( O k ), and thus p Ad( x e ) ( X ) ∈ O k [ X ].Hence Ad( x e ) has integral eigenvalues, and thus the eigenvalues of Ad( x ) are algebraic in-tegers. The coefficients of the characteristic polynomial are symmetric polynomials on theeigenvalues (2.3), thus the coefficients of p Ad( x ) ( X ) are algebraic integers. Since G is adjoint, x ∈ G( k ) [2, Proposition 1.2], so Ad( x ) ∈ SL d ( k ) hence p Ad( x ) ( X ) ∈ k [ X ], and we conclude p Ad( x ) ( X ) ∈ O k [ X ]. niversal Systole Bounds 9 Next we show that if y ∈ SL d ( k ) is semisimple such that p y ( X ) ∈ O k [ X ], then p ι ( y ) ( X ) ∈ Z [ X ]. To begin, since y ∈ SL d ( k ), it follows that ι ( y ) ∈ SL d d ( Q ) and hence p ι ( y ) ( X ) hasrational coefficients. Meanwhile ι , restricted to SL d ( k ), gives the diagonal embeddingSL d ( k ) → Y σ ∈ V ∞ SL d ( σ ( k )) ⊂ SL d d ( L )where V ∞ is the set of all real embeddings of k and L is the Galois closure of k . Since ι ( y )and ι ( y ) are GL d d ( L )-conjugate by Lemma 3.1, they have equal characteristic polynomials.If p y ( X ) = P ni =0 c i X i for c i ∈ O k , then it follows that p ι ( y ) ( X ) = Y σ ∈ V ∞ n X i =0 σ ( c i ) X i ! . Since each σ ( c i ) is an algebraic integer, all of the coefficients of p ι ( y ) ( X ) are algebraic integers.We deduce that the coefficients of p ι ( y ) ( X ) are both rational and algebraic integers, thusconcluding p ι ( y ) ( X ) ∈ Z [ X ]. (cid:3) While there are many reasonable Riemannian metrics to put on X n := SO( n ) \ SL n ( R ), asin [10, Equation (10)], we fix once and for all the Riemannian metric on X n given by(3.1) h X, Y i := 2 tr( XY ) , X, Y ∈ sl n ( R ) . This has the nice geometrical property that the inclusions of the hyperbolic plane H → X n (associated to any 2 × ( R ) → SL n ( R )) are totally geodesic,isometrical immersions (i.e., the images have constant curvature -1).For any inclusion of a connected semisimple lie group H → SL n ( R ), this induces a subspacemetric (which we will denote g ) on the corresponding symmetric space ( H ∩ SO( n )) \ H ⊂ X n . Proof of Theorem B. Let Γ ⊂ G( R ) be an arithmetic subgroup and N := K \ G( R ) / Γ, where K is a maximal compact subgroup of G( R ). By [10, Theorem 7.2], we have an embeddingof Lie groups ρ : G( R ) → SL n ( R ) where n = | k : Q | dim G. It follows from Proposition 3.2that if x ∈ Γ is semisimple, then p ρ ( x ) ( X ) ∈ Z [ X ], and hence by our Mahler-Length TheoremA and Proposition 2.1, we have(3.2) sys( N, g ) ≥ √ √ n f ( n ) , where sys( N, g ) denotes the systole of N relative to the subspace metric (3.1), and f ( n ) isas in (2.2). (cid:3) Large Systoles and the Proof of Theorem C Let N be a compact, simple arithmetic manifold. In [10, Theorem B] we showed that N has a finite degree cover N ′ whose systole satisfies sys( N ′ ) ≥ c log(Vol( N ′ )) + c for certainpositive constants c and c . In order to prove Theorem C it therefore suffices to derivean upper bound for the systole of N ′ whose order of magnitude is also log(Vol( N ′ )). Thisfollows from the following proposition. Proposition 4.1. Let X be a symmetric space of noncompact type and Γ be a discrete, cocom-pact subgroup of isometries of X having finite covolume. Then sys( X / Γ) ≤ c log(Vol( X / Γ)) for some positive constant c depending only on X . Proof. Let B be a closed ball of radius R in X which has volume greater than Vol( X / Γ).The projection π : B → X / Γ cannot be one-to-one, hence there exist distinct points x , x ∈ B such that π ( x ) = π ( x ). Then x = γx for some semisimple element γ ∈ Γ, andconsequently the translation length ℓ ( γ ) of γ satisfies ℓ ( γ ) ≤ R . The proposition now followsfrom [9, Theorem A], where it is shown that the volume of B is asymptotically equivalentto R r − e hR as R tends to infinity, where r is the rank of X and h > X . Indeed, the result of [9, Theorem A] shows that the volume of B is greaterthan c e hR for all R ≥ R , where c and R are positive constants. In particular the volumeof B is will be greater than Vol( X / Γ) for any R ≥ R satisfying R > h log(Vol( X / Γ) /c ),and there clearly exists a positive constant c for which such an R may be taken to satisfy R < c log(Vol( X / Γ)). (cid:3) Geometry of Special Linear Orbifolds and Theorem D For n ≥ 2, we fix the globally symmetric space X n := SO( n ) \ SL n ( R ). If Γ ⊂ SL n ( R ) is alattice, then X n / Γ is a special linear orbifold . In this section, we give a brief overview of theconstruction and geometry of what we call standard special linear orbifolds. For a detailedconstruction, we refer the reader to [10, Section 4]. This construction requires knowledge ofthe theory of central simple algebras, for which we refer the reader to [19].Let A be a central simple algebra over Q of dimension n ≥ 4. The Wedderburn Struc-ture Theorem states that A ∼ = M m ( D ) for some division algebra D over Q where n = m (dim D ) .Let ℓ denote either R or the field of p -adic numbers for some rational prime p . Then A ⊗ Q ℓ isa central simple algebra over ℓ , and again by the Wedderburn Structure Theorem, there existsa division algebra D ℓ over ℓ such that there exists an isomorphism h : A ⊗ Q ℓ ∼ = M m ′ ( D ℓ ). Ifdim D ℓ > 1, we say A ramifies over ℓ , otherwise we say A is unramified over ℓ .A Z - order in A is a subring O which is also a finitely generated Z -module for which A = Q ⊗ Z O . An order is maximal if it is maximal with respect to inclusion. Maximal ordersalways exist [19, 10.4]. Let O be a maximal order.To produce our standard special linear manifolds, we fix a central simple algebra A over Q that is unramified over R . Fix the isomorphism h : A ⊗ Q R → M n ( R ). Via that isomorphism,we can define the norm N ( a ) := det( h ( a ⊗ h ).Let O denote the set of norm one elements of O . Following the convention of [14, 8.3.4], ifΓ ⊂ h ( O ), then we say Γ is a standard arithmetic lattice in SL n ( R ) derived from a centralsimple algebra and similarly M := X n / Γ is a standard special linear orbifold derived froma central simple algebra . Any orbifold commensurable to such an M is a standard linearorbifold .As a locally symmetric space, M has a one parameter family of Haar measures induced from X n . We choose the normalization (3.1) which we called the geometric metric [10, Section2], in which the natural embedding SL ( R ) → SL n ( R ) as a 2 × H → X n . Proof of Theorem D. Let M be a special linear orbifold derived from a central simple alge-bra., i.e. M ∼ = X n / Γ where Γ ⊂ O , the norm 1 elements of a maximal order O . Let x ∈ Γbe semisimple and let p x ( X ) denote the characteristic polynomial of h ( x ⊗ 1) for h as above. niversal Systole Bounds 11 It is a fundamental result [19, Theorem 8.6] that p x ( X ) is a monic, degree n polynomialin Z [ X ]. (Note that p x need not be irreducible.) By our Mahler-Length Theorem A andProposition 2.1, we have then sys( M ) ≥ √ √ n f ( n ) , where f ( n ) is as in (2.2), and the resultfollows. (cid:3) Proof of Theorem E Let p > k be a totally real number field with ring of integers O k and p , . . . p s be primes of k .Let B be the division algebra over k with Hasse invariant p at each of the primes p , . . . , p s − and Hasse invariant ap at the prime p s where a is a positive integer satisfying a + ( s − ≡ p ). Since the sum of these Hasse invariants is an integer, it follows from the Albert-Brauer-Hasse-Noether theorem that B exists and has degree p over k (see [12, Theorem2.1].Let O be a maximal order of B and O denote the multiplicative subgroup of O ∗ consistingof the elements of O ∗ having reduced norm one. Since B ⊗ Q R ∼ = M p ( R ) [ k : Q ] we obtainan embedding B ֒ → SL p ( R ) [ k : Q ] . Let Γ denote the image in SL p ( R ) [ k : Q ] of O under thisembedding. Let X denote the symmetric space of SL p ( R ) [ k : Q ] and define M := X / Γ to bethe arithmetic orbifold associated to Γ.Let γ ∈ Γ be a semisimple element whose translation length ℓ ( γ ) realizes the systole of M .As a consequence of Proposition 4.1 we have the inequality(6.1) ℓ ( γ ) ≤ c log(Vol( X / Γ))for some positive constant c depending only on X (and thus only on p and the degree of k ).We now obtain an upper bound for Vol( X / Γ) using [11, Proposition 3.2]:(6.2) Vol( X / Γ) ≤ c d p − k s Y i =1 N k/ Q ( p i ) ! p ( p − for some positive constant c which is explicitly given in [11, Proposition 3.2] and dependsonly on p and the degree of k . (That the dependence only depends on the degree of k followsfrom the trivial bound ζ k ( s ) ≤ ζ ( s ) [ k : Q ] .)We now need a lower bound for ℓ ( γ ). To do so we will employ the bound given in TheoremA. There is a potential issue however. The hypothesis of Theorem A is that the semisimpleelement γ lies in SL n ( R ) for some positive integer. Our element γ , on the other hand, lies inSL p ( R ) [ k : Q ] . To resolve this we use the fact that SL p ( R ) [ k : Q ] is a Lie subgroup of SL p [ k : Q ] ( R )and apply Theorem A with n = p [ k : Q ]:(6.3) ℓ ( γ ) ≥ s p [ k : Q ] log( M ( p γ )) . Let L = k ( γ ) be the extension of k generated by adjoining to k a preimage in B of γ . Thisextension has degree p and, by Theorem 32.15 of [19], for every prime P of L lying abovea prime p ∈ { p , . . . , p s } , satisfies [ L P : k p ] = p . In particular none of the primes p i splitcompletely in L/k .In order to relate the relative discriminant ∆ L/k of L over k we will make use of a result ofSilverman [21, Theorem 2]:(6.4) log( M ( p γ )) ≥ p ( p − 1) log( N k/ Q (∆ L/k )) − p log( p )2( p − . By combining (6.3) and (6.4) we obtain positive constants c , c depending only on p andthe degree of k such that(6.5) ℓ ( γ ) ≥ c + c log( N k/ Q (∆ L/k )) . Finally, by combining (6.1), (6.2) and (6.5) we obtain positive constants c , c , c such that N k/ Q (∆ L/k ) ≤ c d c k s Y i =1 N k/ Q ( p i ) ! c , which finishes the proof of Theorem E. References [1] A. Borel Linear Algebraic Groups . Graduate Texts in Mathematics, , Springer-Verlag, (1991).[2] A. Borel, G. Prasad. Finiteness theorems for discrete subgroups of bounded covolume in semi-simplegroups . Inst. Hautes ´Etudes Sci. Publ. Math. No. 69 (1989), 119–171.[3] D. W. Boyd. Reciprocal polynomials having small measure . Math. Comp. 35 (1980), no. 152, 1361–1377.[4] D. W. Boyd. Reciprocal polynomials having small measure. II . Math. Comp. 53 (1989), no. 187, 355–357,S1–S5.[5] V. Emery, J. G. Ratcliffe, and S. T. Tschantz. Salem numbers and arithmetic hyperbolic groups . Trans.Amer. Math. Soc. 372 (2019), no. 1, 329–355.[6] E. 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