Conformal fractals for normal subgroups of free groups (Q2922920)

This article is about the multifractal analysis of conformal graph directed Markov systems (GDMS) \(\Phi\) associated to the free group \(F_d\) with \(d \geq 2\) generators. The limit set of the GDMS is decomposed along radial limit sets corresponding to non-trivial normal subgroups \(N\).NEWLINENEWLINEA GDMS \((N,\Phi)\), called the \(N\)-induced GDMS of \(\Phi\) (compare Definition 3.9), is associated to the normal subgroup \(N\) in the following way. For the whole system \(\Phi\), the vertices are the generators of \(F_d\) and their inverses. The set of vertices of the system \((N,\Phi)\) then consists of the products of the generators of \(F_d\) which belong to \(N\) and which are minimal in the sense that no contiguous subproduct is an element of \(N\).NEWLINENEWLINEGiven a normal subgroup \(N\) of \(F_d\), the symbolic radial limit set \(\Lambda_r (N)\) is defined in Definition 3.8 as the set of infinite words \(w\) for which there is a element \(g \in F_d\) such that for infinitely many \(n\), the product \(w_1 \cdot \dots \cdot w_n\) is an element of the coset \(N g\). The symbolic uniform radial limit set \(\Lambda_{ur} (N)\) is a stronger version of the above definition stating that the orbit visits only finitely many different cosets.NEWLINENEWLINEThe first main results is Theorem 1.1. It states the following: {\parindent=0.6cm \begin{itemize}\item[--] The Hausdorff dimensions of the radial and uniformly limit sets of \(N\) (that is the projections of the corresponding symbolic sets) are both equal to the exponent of convergence of some Poincaré series. \item[--] If the quotient \(F_d/N\) is non-amenable, then NEWLINE\[NEWLINE\dim_H (\pi_\Phi(\Lambda_r (N)))<\dim_H (\pi_\Phi(\Lambda_r (F_d)))NEWLINE\]NEWLINE (\(\dim_H\) denotes the Hausdorff dimension, \(\pi_\Phi\) is the projection from the set of words to the limit set). \item[--] Assume \(N\) is an ``asymptotic symmetry'', that is, it satisfies some kind of weak asymptotic comparison between Poincaré series associated to the coset \(N g\) and to the coset \(N g^{-1}\), see Definition 4.1. Then, \item[--] If \(F_d / N\) is amenable, then \(\dim_H (\pi_\Phi(\Lambda_r (N)))=\dim_H (\pi_\Phi(\Lambda_r (F_d)))\). \item[--] In any case, the dimension of the radial limit set for \(N\) is at least half of the dimension of the radial limit set for \(F_d\). The inequality is strict whenever \(N\) is ``symmetric'' (a stronger relation between the Poincaré series, see Definition 4.1). \item[--] Assume the system \((N, \Phi)\) is of divergence type. Then, \(F_d / N\) is amenable. \item[--] The equality \(\dim_H (\pi_\Phi(\Lambda_r (N)))=\dim_H (\pi_\Phi(\Lambda_r (F_d)))\) is equivalent to \((N, \Phi)\) being ``symmetric on average'' (see Definition 4.1). The second main result, Theorem 1.2, is about the multifractal analysis of such systems. It gives similar (but not entirely equivalent) results as Theorem 1.1 for the multifractal level sets. NEWLINENEWLINE\end{itemize}} Those theorems are then applied to the study of the Lyapunov spectrum for normal subgroups of Kleinian groups of Schottky type. This is the object of Corollary 1.5. Its statement is as follows.NEWLINENEWLINELet \(\Gamma\) be a Kleinian group of Schottky type and let \(N\) be a non-trivial normal subgroup of \(\Gamma\). For (an admissible) \(\alpha\), let \(\mathcal{L} (\alpha)\) be the multifractal level set of the limit set of \(\Gamma\) and \(\mathcal{L}_N (\alpha)\) the corresponding level set for \(N\). Then, NEWLINE\[NEWLINE \frac{\dim_H (\mathcal{L} (\alpha))}{2} < \dim_H (\mathcal{L}_N (\alpha)) \leq \dim_H (\mathcal{L} (\alpha)). NEWLINE\]NEWLINE Moreover, the second inequality is an equality if and only if \(\Gamma/N\) is amenable.NEWLINENEWLINEFinally, if the system associated to \(N\) is of divergence type for some exponent then the quotient \(\Gamma / N\) is amenable.