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Asif Shaikh

Publications and source records attributed to Asif Shaikh.

5 recordsLinked to original sources
arXiv2024

The cogrowth inequality from Whitehead's algorithm

This article focuses on free factors H <= F_m of the free group F_m with finite rank m > 2, and specifically addresses the implications of Ascari's refinement of the Whitehead automorphism phi for H as introduced in \cite{ascari2021fine}. Ascari showed that if the core Delta_H of H has more than one vertex, then the core Delta_{phi(H)} of phi(H) can be derived from Delta_H. We consider the regular language L_H of reduced words from F_m representing elements of H, and employ the construction of mathcal{B}_H described in \cite{DGS2021}. mathcal{B}_H is a finite ergodic, deterministic automaton that recognizes L_H. Extending Ascari's result, we show that for the aforementioned free factors H of F_m, the automaton mathcal{B}_{phi(H)} can be obtained from mathcal{B}_H. Further, we present a method for deriving the adjacency matrix of the transition graph of mathcal{B}_{phi(H)} from that of mathcal{B}_H and establish that alpha_H < alpha_{phi(H)}, where alpha_H, alpha_{phi(H)}$ represent the cogrowths of H and phi(H), respectively, with respect to a fixed basis X of F_m. The proof is based on the Perron-Frobenius theory for non-negative matrices.

math.GR↗
arXiv2023

Multivariate growth and cogrowth

We investigate a multivariate growth series $Γ_L({\bf z}), {\bf z} \in \mathbb{C}^d$ associated with a regular language $L$ over an alphabet of cardinality $d.$ Our focus is on languages coming from subgroups of the free group and from subshifts of finite type. We develop a mechanism for computing the rate of growth $φ_L({\bf r})$ of $L$ in the direction ${\bf r} \in \mathbb{R}^d$. Using the concave growth condition (CG) introduced by the second author in \cite{quint2002divergence} and the results of Convex Analysis we represent $ψ_L({\bf r}) = \log\left(φ_L({\bf r})\right)$ as a support function of a convex set that is a closure of the $\textrm{Relog}$ image of the domain of absolute convergence of $Γ_L({\bf z})$. This allows us to compute $ψ_L({\bf r})$ in some important cases, like a Fibonacci language or a language of freely reduced words representing elements of a free group $F_2$. Also we show that the methods of the Large deviation theory can be used as an alternative approach. Finally, we suggest some open problems directed on the possibility of extensions of the results of the first author from \cite{grigorchuk1980symmetrical} on multivariate cogrowth.

math.GR↗
arXiv2021

Finitely generated subgroups of free groups as formal languages and their cogrowth

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

math.GR↗
arXiv2016

Zeta functions of finite Schreier graphs and their zig zag products

We investigate the Ihara zeta functions of finite Schreier graphs $Γ_n$ of the Basilica group. We show that $Γ_{1+n}$ is $2$ sheeted unramified normal covering of $Γ_n, ~\forall~ n \geq 1$ with Galois group $\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}.$ In fact, for any $n > 1, r \geq 1$ the graph $Γ_{n+r}$ is $2^n$ sheeted unramified, non normal covering of $Γ_r.$ In order to do this we give the definition of the $generalized$ $replacement$ $product$ of Schreier graphs. We also show the corresponding results in zig zag product of Schreier graphs $Γ_n$ with a $4$ cycle.

math.GR↗