Let $S\_1, \ldots, S\_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota\_i$ be the inversion in $S\_i$. Viewing the conformal group of $\mathbb{S}^{n-2} = \mathbb{R}^{n-2} \cup \{\infty\}$ as an index-2 subgroup of $\mathrm{O}(n-1,1)$ yields a representation of the product $\iota\_1\ldots\iota\_n$ as an $n \times n$ real matrix whose characteristic polynomial has integer coefficients. Since $\iota\_1\ldots\iota\_n$ maps the closure of the exterior of $S\_n$ into the exterior of $S\_n$ (indeed, into the interior of $S\_1$), this matrix has a unique eigenvalue $\lambda\_n$ of modulus larger than $1$. (The positive algebraic integer $\lambda\_n$ is independent of all the choices made above.)


I was wondering what is known about the numbers $\lambda\_n$. For instance, how does this sequence grow? Is the degree of $\lambda\_n$ precisely $n$ for $n$ even, and $n-1$ for $n$ odd? A positive answer to the latter question would imply that the $\lambda\_n$ are Salem numbers for all $n \geq 4$.



What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?
I computed the matrices for $n=3,...,10$ in Mathematica and found the factorizations of the characteristic polynomials:


$$-(1 + x) (1 - 18 x + x^2),$$
$$ 1 - 68 x - 122 x^2 - 68 x^3 + 
 x^4,$$ $$-(1 + x) (1 - 228 x - 314 x^2 - 228 x^3 + x^4),$$
$$1 - 710 x - 2033 x^2 - 2708 x^3 - 2033 x^4 - 710 x^5 + 
 x^6,$$ $$ -(1 + x) (1 - 2166 x - 4913 x^2 - 6324 x^3 - 4913 x^4 - 
 2166 x^5 + x^6),$$
$$1 - 6536 x - 23780 x^2 - 42680 x^3 - 50618 x^4 - 42680 x^5 - 
 23780 x^6 - 6536 x^7 + 
 x^8,$$ $$-(1 + x) (1 - 19656 x - 58724 x^2 - 96120 x^3 - 109754 x^4 - 
 96120 x^5 - 58724 x^6 - 19656 x^7 + x^8),$$
$$1 - 59018 x - 255443 x^2 - 547448 x^3 - 792878 x^4 - 884732 x^5 - 
 792878 x^6 - 547448 x^7 - 255443 x^8 - 59018 x^9 + x^{10}$$


This seems to obey the pattern you seek. I’ve also checked this pattern up to $n=50$.


The maximal eigenvalues are computed to be
$$\{17.9443, 69.7628, 229.373, 712.857, 2168.27, 6539.64, 19659., \
59022.3\}$$


**Addendum:** Here is how I did the computation. In the Lorentzian model of hyperbolic space in $\mathbb{R}^{n-1,1}$, the $n$ planes intersecting $S^{n-2} \subset \mathbb{RP}^{n-1}$ in $n$ tangent spheres will be orthogonal to vectors $e\_i$ such that $\langle e\_i, e\_i\rangle =1$ in the Lorentzian product (here the hyperboloids defining hyperbolic space will be $\{x | \langle x, x\rangle =-1\}$). The tangency between the spheres corresponds to $\langle e\_i , e\_j\rangle =-1, i\neq j$ (see [Section 1.3 of this paper](https://link.springer.com/article/10.1007/s10711-017-0280-7)).


Reflection through $e\_i^{\perp}$ is given by $R\_i(x)= x- 2\langle x, e\_i\rangle e\_i$, and thus we see $R\_i(e\_i)=-e\_i, R\_i(e\_j) = e\_j+2e\_i, j\neq i$. From these formulas we can write down matrices and compute the products $R\_1\cdots R\_n$, which is how I derived these computations in Mathematica.


One might be able to guess/ prove an inductive formula for the polynomials which then could yield the possibility of applying an irreducibility criterion. I can already see formulae for some of the coefficients of the matrices which might allow one to guess a formula for the characteristic polynomial.


