—   dimension (an integer with $3\leq n\leq9$)

—   Coxeter symbol (a Coxeter symbol of a finite-volume hyperbolic Coxeter simplex)

For $W(t)=\sum_{k\geq0}a_k t^k$, Steinberg's formula [3] gives $1/W(t^{-1})=\sum_{T\subseteq S,\ W_T\text{ finite}} (-1)^{|T|}/W_T(t)$, where $W_T$ is the subgroup generated by $T$ and $W_T(t)=\prod_i(1+t+\cdots+t^{d_i-1})$ is its growth polynomial, with $d_i$ the degrees of $W_T$ when $W_T$ is finite.

The radius of convergence of $W(t)$ is the smallest positive pole of the reduced rational function $W(t)$, and $\tau$ is its reciprocal.

The index-$2$ orientation-preserving subgroup has a different word metric, so its growth rate is a different number and is not held here.

Entries are labelled by Coxeter symbols [7]: $[p,q,r]$ is the linear diagram with branch labels $p,q,r$, $[(p,q,r,s)]$ is the cyclic diagram, and $[5,3^{1,1}]$ is the branched diagram with one edge labelled $5$ before a split into two edges labelled $3$.

Among all cocompact hyperbolic Coxeter groups in $\mathbb{H}^3$, $[3,5,3]$ has the smallest growth rate, and it is the only group attaining it [2].

Every value in this table is a Salem number: each minimal polynomial below is reciprocal, with one root outside the unit circle and the rest on it or inside. That is a property of the *cocompact* groups, which are the nine held here, and not of hyperbolic Coxeter simplex groups in general [4]. The finite-volume non-cocompact tetrahedral groups, which this table does not hold, have growth rates that are Perron and often Pisot rather than Salem: $[6,3,6]$ gives the golden ratio, a root of $t^2-t-1$, and $[4,3,6]$ the plastic number, a root of $t^3-t^2-1$.

Sage

def growth_rate(M):
    # M is a Coxeter matrix; the final line uses [3,5,3].
    t = polygen(QQ, 't')
    total = 0
    for T in Subsets(range(len(M))):
        T = sorted(T)
        C = CoxeterMatrix([[M[i][j] for j in T] for i in T]) if T else None
        if T and not C.is_finite():
            continue
        W_T = prod(sum(t^k for k in range(d)) for d in CoxeterGroup(C).degrees()) if T else 1
        total += (-1)^len(T) / W_T
    q = (1 / total(1/t)).denominator()
    return max(q.roots(AA, multiplicities=False)).n(digits=100)

growth_rate([[1,3,2,2], [3,1,5,2], [2,5,1,3], [2,2,3,1]])

James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, 1990.

Ruth Kellerhals and Alexander Kolpakov, The minimal growth rate of cocompact Coxeter groups in hyperbolic 3-space, Canad. J. Math. 66 (2014), no. 2, 354-372. (doi)

Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society 80, 1968.

W. Parry, "Growth series of Coxeter groups and Salem numbers", Journal of Algebra 154 (1993), 406-415. (doi)

Wikipedia: Coxeter group

Wikipedia: Growth rate (group theory)

Wikipedia: Coxeter notation

Growth rates of hyperbolic Coxeter triangle groups —   the same growth rates for the simplex groups of dimension $2$, the hyperbolic Coxeter triangles

Viswanath's constant —   another exponential growth rate, from random Fibonacci recurrences rather than Coxeter groups

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds all nine compact Coxeter tetrahedral groups in $\mathbb{H}^3$, the complete compact rank-$4$ list in Humphreys [1], and no paracompact tetrahedral groups or simplex groups in dimensions $4$ to $9$)

How they were obtained:

The generator computes the rational growth series exactly from (1), using the finite special subgroups of each Coxeter diagram. It factors the reciprocal denominator, isolates the largest real root greater than $1$ in interval arithmetic, and writes $100$ digits.

more

As checks, the minimal polynomial printed for $[3,5,3]$ was reproduced exactly, the Coxeter relations were checked in the exact Tits representation, and the first growth-series coefficients of every row were compared with breadth-first counts in that representation.