Growth rates of Löbell reflection groups

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Numbers

$n$

$\tau_n$

7.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673518

comment: The regular right-angled hyperbolic dodecahedron has $\tau_5=4+\sqrt{15}$.

9.898979485566356196394568149411782783931894961313340256865385134501920754914630053079718866209280470

11.91607978309961604256732829156161704841550123079434032287971966914282245910565303676575252718310918

13.92820323027550917410978536602348946777122101524152251222322791780773206763520014832458474702899430

15.93725393319377177150484726091778127713077754924735054110500337760320646969085088328117865942363083

10:

17.94427190999915878563669467492510494176247343844610289708358898164208370255121959765765763351512910

11:

19.94987437106619954734479821001206005178126563676806079117604643834945392782713154012653019738487195

12:

21.95445115010332226913939565601604267905489389995966508453788899464986554245445467601716872327741252

13:

23.95826074310139802112984075619561661399108628513719791215803151415392495069736590783473293461878122

14:

25.96148139681572046193193487217599331541040861411669309942270879561923475568808874280072181321122047

15:

27.96424004376894116988392454108637602064017972392484401020358665726159025113189426406779308178955746

16:

29.96662954709576554233499492926619720702407923111490778521498186928014062522775611190723958077751829

17:

31.96871942267131199907024517698061384156734970437542667323683764606239452587634292284656231142583919

18:

33.97056274847714058562026469051637694283606250452337687812015685588878974154528446620465041193169887

19:

35.97220075561142845760032673419901616976638598195453222686629524904424975078242640607381976419083281

20:

37.97366596101027599199336126659631120231733083595130096114502911675556663183542932806548865027580177

21:

39.97498435543817891578038232805828719786257611664143653778310303067737567062314451275144516054342469

22:

41.97617696340303093982907027359875196950543715363007969751511527160011851002201382838770577866388836

23:

43.97726097583591056720163586089555091651592577019931656179808798814194865062082108070611677416772426

24:

45.97825058615211463940244587287571727288105783193116947079950988226360288582961850810850146238715249

25:

47.97915761656359770798719032081346959998353520952064673242654557224128617953732041246095723218459430

26:

49.97999199359359282338757248375917784429183991196662523381188772243844480233405800253334444888536235

27:

51.98076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

28:

53.98147512646408293110061122299108994491073620043502270883859901592000288094706183071381966861132006

29:

55.98213715926644513666342662696167094982872208257496401316859189534035364642174903524266168873564605

30:

57.98275349237887714743732831433954344628026574951794617739184961423628874530736084342512638400723500

Definition

Listed is the exponential growth rate $\tau_n$ of the group $W_n$ generated by reflections in the faces of the compact right-angled Löbell polyhedron $L_n$ [1], using those face reflections as generators [2].

Parameters

$n$

— Number of sides of either base (an integer with $n\geq5$)

Formulas

(1)

Let $S_n$ be the set of face reflections and $\ell_{S_n}(w)$ the minimum word length of $w\in W_n$ in $S_n$. Then $a_{n,k}=\#\{w\in W_n:\ell_{S_n}(w)=k\}$ and $\tau_n=\limsup_{k\to\infty}a_{n,k}^{1/k}$.

(2)

For the growth series $F_n(t)=\sum_{k\geq0}a_{n,k}t^k$, with $a_{n,0}=1$, equation (4.3) of [2] gives $F_n(t)=\frac{(1+t)^3}{1-(2n-1)t+(2n-1)t^2-t^3}=\frac{(1+t)^3}{(1-t)(1-(2n-2)t+t^2)}$.

(3)

$\tau_n=n-1+\sqrt{n(n-2)}$ is the larger root of $x^2-(2n-2)x+1=0$. The square root is positive, and the radius of convergence of $F_n(t)$ is $\tau_n^{-1}$.

(4)

$\log\tau_n=\operatorname{arcosh}(n-1)$, where $\operatorname{arcosh}$ is the nonnegative real inverse of $\cosh$.

(5)

$\tau_{10}=9+4\sqrt5=((1+\sqrt5)/2)^6$, so $\log\tau_{10}=6R_{\mathbb{Q}(\sqrt5)}$, where $R_{\mathbb{Q}(\sqrt5)}$ is the regulator of the field of fundamental discriminant $5$.

Comments

(6)

A pentagonal flower consists of an $n$-gon and $n$ surrounding pentagons, each adjacent to that base and to its two cyclic neighbours. Gluing two such flowers along their boundaries to give a trivalent sphere defines $L_n$ [1], Section 4. Its hyperbolic realization has all dihedral angles $\pi/2$ and is unique up to isometry. It has $2n+2$ faces, $6n$ edges and $4n$ vertices. We use sectional curvature $-1$, as in the companion table of volumes.

(7)

The generating set $S_n$ contains all $2n+2$ face reflections. The coefficient $a_{n,k}$ counts distinct group elements at minimum length $k$, not all words or all geodesic words. The same generating-set convention is used for the growth rates of hyperbolic Coxeter simplex groups.

(8)

The number $\tau_n$ is an irrational quadratic unit of norm $1$ in $K_n=\mathbb{Q}(\sqrt{n(n-2)})$, with conjugate $\tau_n^{-1}$. Thus $\log\tau_n$ is a positive integer multiple of the regulator of $K_n$; the unit $\tau_n$ need not be fundamental.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField
R = RealBallField(numberdb.bits(100, losing=64))
def growth_rate(n):
    n = ZZ(n)
    if n < 5:
        raise ValueError('Use n >= 5')
    return R(n-1) + R(n*(n-2)).sqrt()
[growth_rate(n) for n in range(5, 31)]

References

[1]

Taiyo Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebraic & Geometric Topology 8 (2008), 1523–1565. Section 4. (arXiv) (doi)

Links

[2]

Ruth Kellerhals and Alexander Kolpakov, The minimal growth rate of cocompact Coxeter groups in hyperbolic 3-space, Sections 2.2 and 4.2

Similar tables

Volumes of right-angled hyperbolic antiprisms and Löbell polyhedra — The $L_n$ rows give volumes of the same compact polyhedra with the same base index $n$; each volume is the covolume of its full face-reflection group.

Growth rates of hyperbolic Coxeter simplex groups — Growth rates in the face reflections for simplex chambers, including the nine compact tetrahedra.

Regulators of real quadratic fields — The logarithm of $\tau_n$ is an integral multiple of the regulator of $\mathbb{Q}(\sqrt{n(n-2)})$.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every $L_n$ with $5\leq n\leq30$, covering consecutive small and moderate combinatorial sizes from the dodecahedron through 62 faces and 120 vertices; the practical, noncanonical cutoff of 30 matches the compact objects in the companion volume table)

How they were obtained:

The generator uses Arb real balls at 397 bits (100 decimal digits plus 64 guard bits) to evaluate the positive radical in (3). Every ball is finite and has relative radius smaller than $2^{-395}$; every stored 100-digit decimal interval contains its computed ball.

For all 26 entries, a separate checker constructs the face-adjacency graph, enumerates its finite special subgroups, and obtains (2) by Steinberg's formula [2], equation (2.5). Exact rational bisection of the reciprocal denominator supplies an independent root bracket inside each written decimal interval. Direct group-element enumeration through length three checks the first growth coefficients, with finite elementary abelian and free-product control groups. The quadratic, reciprocal trace, logarithm and exact fundamental-unit power identities are checked for every entry; stored regulators in the companion table provide additional comparisons where its discriminant range overlaps. The attached checks run both on generated values and on values read back from the database.