The growth rates of ideal Coxeter polyhedra in hyperbolic 3-space

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1 The growth rates of ideal Coxeter polyhedra in hyperbolic 3-space Jun Nonaka Waseda University Senior High School Joint work with Ruth Kellerhals (University of Fribourg) June Boston University / Keio University Workshop 2017 Joint work with Ruth Kellerhals (University of1 Fribo / 21

2 Coxeter polyhedron in H n B n := {x ( R n+1 x < 1} ( ) ) 2 H n := B n, 2 n 1 x 2 i=1 dx i 2 hyperbolic n-space Definition A Coxeter polyhedron in H n is an n-dimensional convex polyhedron in H n with its dihedral angles are submultiples of π. A convex polyhedron in H n is compact if H n is empty, and is ideal if all of its vertices are in H n. If a convex polyhedron with finite volume is non-compact, then it has vertices in H n finitely. We call these vertices cusps. Joint work with Ruth Kellerhals (University of2 Fribo / 21

3 Coxeter polyhedron in H n Examples of Coxeter polyhedra in hyperbolic spaces Joint work with Ruth Kellerhals (University of3 Fribo / 21

4 Hyperbolic Coxeter group Definition (G, S) is a Coxeter system if G = s S (st) m s,t = 1 for s, t S with satisfying the following conditions 1. m s,t = m t,s 2. s S, then m s,s = 1 3. if s t, then m s,t {2, 3,, + } G is called Coxeter group. A group generated by reflections obtained by each face of a Coxeter polyhedron (in H n ) is a (n-dimensional hyperbolic) Coxeter group. Joint work with Ruth Kellerhals (University of4 Fribo / 21

5 Hyperbolic Coxeter group Coxeter polygons in H 2 G 1 = g 1, g 2, g 3 g 2 1 = g 2 2 = g 2 3 = (g 1g 2 ) 2 = (g 2 g 3 ) 4 = 1 G 2 = g 1, g 2, g 3, g 4 g 2 1 = g 2 2 = g 2 3 = g 2 4 = 1 Joint work with Ruth Kellerhals (University of5 Fribo / 21

6 Hyperbolic Coxeter group Coxeter polygons in H 2 G 1 = g 1, g 2, g 3 g 2 1 = g 2 2 = g 2 3 = (g 1g 2 ) 2 = (g 2 g 3 ) 4 = 1 G 2 = g 1, g 2, g 3, g 4 g 2 1 = g 2 2 = g 2 3 = g 2 4 = 1 Joint work with Ruth Kellerhals (University of6 Fribo / 21

7 Hyperbolic Coxeter group Coxeter polygons in H 2 G 1 = g 1, g 2, g 3 g 2 1 = g 2 2 = g 2 3 = (g 1g 2 ) 2 = (g 2 g 3 ) 4 = 1 G 2 = g 1, g 2, g 3, g 4 g 2 1 = g 2 2 = g 2 3 = g 2 4 = 1 Joint work with Ruth Kellerhals (University of7 Fribo / 21

8 Hyperbolic Coxeter group Coxeter polygons in H 2 G 1 = g 1, g 2, g 3 g 2 1 = g 2 2 = g 2 3 = (g 1g 2 ) 2 = (g 2 g 3 ) 4 = 1 G 2 = g 1, g 2, g 3, g 4 g 2 1 = g 2 2 = g 2 3 = g 2 4 = 1 Joint work with Ruth Kellerhals (University of8 Fribo / 21

9 Hyperbolic Coxeter group The tessellations of H 2 Reference. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds Second Edditions, Graduate Texts in Math., 149, Springer (2006) Joint work with Ruth Kellerhals (University of9 Fribo / 21

10 Growth rate G: finitely generated group with generating set S l S (g) := min{n N g = s 1 s n, s i S} (l S (1) = 0) a k := #{g G l S (g) = k} ( a 0 = 1) Definition (Growth function of (G, S)) f S (t) = a k t k Definition (Growth rate of (G, S)) k=0 τ := lim sup k k ak The radius of convergence of f S (t) is 1 τ Joint work with Ruth Kellerhals (University of10 Fribo / 21

11 Previous works Theorem (Cannon-Wagreich 1992, Parry 1993) The growth rates of Coxeter groups obtained from compact Coxeter polyhedra in H n for n = 2, 3 are Salem numbers. A Salem number α > 1 is a real algebraic integer if all its conjugates different from α are in the closed unit disk, and α has at least one conjugate on the unit circle. Joint work with Ruth Kellerhals (University of11 Fribo / 21

12 Previous works Theorem (Floyd 1992) The growth rates of Coxeter groups obtained from non-compact Coxeter polyhedra with finite area in H 2 are Pisot numbers. A Pisot number β > 1 is a real algebraic integer if all algebraic conjugates different from β are of absolute value < 1. Joint work with Ruth Kellerhals (University of12 Fribo / 21

13 Previous works Theorem (Kellerhals-Perren 2011) The growth rates of Coxeter groups obtained from 4-dimensional compact hyperbolic Coxeter polyhedra with at most six 3-faces are Perron numbers. A real algebraic integer γ > 1 is called Perron number if all its conjugates different from γ are of absolute value < γ. Theorem (Komori-Umemoto 2012) The growth rates of Coxeter groups obtained from 3-dimensional hyperbolic Coxeter polyhedra with four or five faces are Perron numbers. Joint work with Ruth Kellerhals (University of13 Fribo / 21

14 Main result 1 Theorem (N-Kellerhals, Komori-Yukita) The growth rate of an ideal Coxeter polyhedron in H 3 is a Perron number. Joint work with Ruth Kellerhals (University of14 Fribo / 21

15 Theorem (Solomon 1966) f S (t): the growth function of an irreducible spherical Coxeter group (Γ, S) is given by f S (t) = l i=1 [m i + 1] where [n] := 1 + t + + t n 1 and {m 1 = 1, m 2,, m l } is the set of exponents (as defined in [Humphreys]). Joint work with Ruth Kellerhals (University of14 Fribo / 21

16 Symbols Coxeter graphs Exponents growth functions A n B n 1, 2,, n [2, 3,, n + 1] 1, 3,, 2n 1 [2, 4,, 2n] D n E 6 E 7 1, 3,, 2n 3, n 1 [2, 4,, 2n 2][n] 1, 4, 5, 7, 8, 11 [2, 5, 6, 8, 9, 12] 1, 5, 7, 9, 11, 13, 17 [2, 6, 8, 10, 12, 14, 18] E 8 F 4 H 3 5 H 4 5 m I 2 (m) 1, 7, 11, 13, 17, 19, 23, 29 [2, 8, 12, 14, 18, 20, 24, 30] 1, 5, 7, 11 [2, 6, 8, 12] 1, 5, 9 [2, 6, 10] 1, 11, 19, 29 [2, 12, 20, 30] 1, m 1 [2, m] [m, n] = [m][n] Table: Exponents and growth functions of irreducible finite Coxeter groups [Kellerhals-Perren] Joint work with Ruth Kellerhals (University of14 Fribo / 21

17 Theorem (Steinberg 1968) G = S R : Coxeter group f S (t) : the growth function of G G T = T R : Coxeter subgroup of G generated by T S f T (t) : growth function of G T F = {T G T is finite} Then 1 f S (t 1 ) = ( 1) T f T (t) T F Joint work with Ruth Kellerhals (University of14 Fribo / 21

18 P : an ideal Coxeter polyhedron in H 3 f P (t) : the growth function of P 1 f P (t 1 ) = 1 f [2] + e 2 [2, 2] + e 3 [2, 3] + e 4 [2, 4] + e 6 [2, 6] = 1 f 1 + t + 6f 2c c (1 + t) 2 + 2c 2f + 2c 1 c (1 + t)(1 + t + t 2 + ) ( 2c 2f c 1 c (1 + t) 2 (1 + t + t 2 )(1 t + t 2 ) f P (t) = [2,2,3](1+t2 )(1 t+t 2 ) (t 1)g P (t) g P (t) = (c 1)t 7 + (c f + 1)t 6 + (c + f c 1 /2 4)t 5 +(2c 2f + (c 1 c 2 )/2 + 2)t 4 + (2f (c 1 c 2 )/2 6)t 3 +(c f + c 1 /2)t 2 + (f 3)t 1 Joint work with Ruth Kellerhals (University of14 Fribo / 21

19 Lemma (Komori-Umemoto 2012) Consider the polynomial g(t) = n b k t k 1 (n 2), k=1 where b k is a non-negative integer. Assume that the greatest common divisor of {k N b k 0} is 1. Then there is a real number r 0, 0 < r 0 < 1 which is the unique zero of g(t) having the smallest absolute value of all zeros of g(t). g P (t) = (c 1)t 7 + (c f + 1)t 6 + (c + f c 1 /2 4)t 5 +(2c 2f + (c 1 c 2 )/2 + 2)t 4 + (2f (c 1 c 2 )/2 6)t 3 +(c f + c 1 /2)t 2 + (f 3)t 1 Joint work with Ruth Kellerhals (University of15 Fribo / 21

20 Main result 2 Ideal Coxeter simplices in H 3 τ(s 1 ) τ(s 2 ) τ(s 3 ) vol(s 1 ) vol(s 2 ) vol(s 3 ) Joint work with Ruth Kellerhals (University of15 Fribo / 21

21 Main result 2 Theorem (N-Kellerhals, Komori-Yukita) The Coxeter tetrahedron S 1 has minimal growth rate among all ideal Coxeter polyhedra of finite volume in H 3, and as such is unique. Its growth rate τ(s 1 ) is the Perron number with minimal polynomial t 5 t 4 t 3 t 2 t 3. Joint work with Ruth Kellerhals (University of16 Fribo / 21

22 P, P : ideal Coxeter polyhedra in H 3 τ(p), τ(p ): the growth rates of P and P If g P (t) g P (t) > 0 for t (0, 1), then 1 τ(p) < 1 τ(p ). Joint work with Ruth Kellerhals (University of17 Fribo / 21

23 Main result 3 τ( ): the growth rate of an ideal Coxeter polyhedron Theorem (N-Kellerhals) Let P and P be ideal Coxeter polyhedra in H 3. Suppose that P has a face F which is isometric to a face F of P, and denote by P F P the ideal polyhedron arising by gluing P to P along their isometric faces F and F. If P F P is a Coxeter polyhedron, then τ(p F P ) > max{τ(p), τ(p )}. Joint work with Ruth Kellerhals (University of17 Fribo / 21

24 τ(s 1 ) τ(s 2 ) τ(s 1 F S 1 ) τ(s 2 F S 2 ) Joint work with Ruth Kellerhals (University of17 Fribo / 21

25 τ(p 1 ) τ(p 2 ) τ(p 1 F P 1 ) = 5 τ(p 2 F P 2 ) Joint work with Ruth Kellerhals (University of18 Fribo / 21

26 The growth rates of Coxeter polyhedra in H 3 Theorem (Yukita) The growth rate of a non-compact Coxeter polyhedron in H 3 is a Perron number. Joint work with Ruth Kellerhals (University of18 Fribo / 21

27 References [1] J.W. Cannon and Ph. Wagreich, Growth functions of surface groups, Math. USSR. Sb., 12 (1971), [2] J.E. Humphreys, Reflection Groups And Coxeter Groups, Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, 29 (1990). [3] R. Kellerhals and G. Perren, On the growth of cocompact hyperbolic Coxeter groups, European J. Combin., 32 (2011), [4] Y. Komori and Y. Umemoto, On the growth of hyperbolic 3-dimensional generalized simplex reflection groups, Proc. Japan Acad., 88 Ser. A (2012), [5] Y. Komori and T. Yukita, On the growth rate of ideal Coxeter groups in hyperbolic 3-space, Preprint, arxiv: [6] Y. Komori and T. Yukita, On the growth rate of ideal Coxeter groups in hyperbolic 3-space, Proc. Japan Acad. Ser. A Math. Sci., 91 (2015), Joint work with Ruth Kellerhals (University of19 Fribo / 21

28 References [7] J. Nonaka, The growth rates of ideal Coxeter polyhedra in hyperbolic 3-space, Preprint, arxiv: [8] J. Nonaka and R. Kellerhals, The growth rates of ideal Coxeter polyhedra in hyperbolic 3-space, to appear in Tokyo Journal of Mathematics. [9] W. Parry, Growth series of Coxeter groups and Salem numbers, J. Algebra 154 no.2 (1993), [10] L. Solomon, The orders of finite Chevalley groups, J. Algebra, 3 (1966), [11] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., 80 (1968). Joint work with Ruth Kellerhals (University of20 Fribo / 21

29 References [12] Y. Umemoto, The growth rates of non-compact 3-dimensional hyperbolic Coxeter tetrahedra and pyramids, proceedings of 19th ICFIDCAA Hiroshima 2011, Tohoku University Press (2013), [13] T. Yukita, On the growth rates of cofinite 3-dimensional Coxeter groups some of whose dihedral angles are of the form π m for m = 2, 3, 4, 5, 6, Preprint, arxiv: [14] T. Yukita, On the growth rates of cofinite 3-dimensional Coxeter groups some of whose dihedral angles are π m for m 7, Preprint, arxiv: v2. Joint work with Ruth Kellerhals (University of21 Fribo / 21

30 Thank you for your attention! Joint work with Ruth Kellerhals (University of21 Fribo / 21