arXiv subjects
Publications and source records attributed to Shrinit Singh.
We present a compact formula for the exact partition function of the $d$-dimensional arithmetic Ising model (AIM). For a $2\times2$ system, we express it analytically using the $q$-Hurwitz-Lerch zeta function and derive explicit forms for the free energy and entropy. Additionally, we find that the entropy increases at high temperatures, supporting the presence of entropic order.
Let $H$ be a finitely generated subgroup of a finitely generated group $F$. Restriction of a uniformly random homomorphism $F\to G$, with $G$ finite, defines a probability measure on $\mathrm{Hom}(H,G)$; for cyclic $H$ this is the usual word measure. We introduce marked and setwise notions of profinite rigidity for these subgroup measures. In a finitely generated LERF group, an isomorphism between finitely generated subgroups preserves all induced measures if and only if its profinite completion extends to an automorphism of the ambient profinite completion. This gives a discrete - profinite orbit criterion and identifies the precise obstruction separating marked from setwise rigidity. For free groups, both notions are invariant under adjoining a free factor. We classify the measure classes of subgroups of $F_n$ containing $[F_n,F_n]$: they are the $\mathrm{Aut}(F_n)$-orbits determined by Smith invariants, and the subgroups determined uniquely by their measures are exactly $F_n^{m}[F_n,F_n]$. Finally, marked rigidity is preserved and reflected by free products compatible with an ambient free-product decomposition. Consequently, for every $n\ge2$ and $1\le r\le n$, there is an infinite-index profinitely rigid subgroup of $F_n$ of rank $r$ whose algebraic closure is $F_n$.
We provide a general structural criterion implying that a group has infinite $m$-almost palindromic width. In particular, we prove that both HNN extensions and free products exhibit infinite $m$-almost palindromic width, with the unique exception of the infinite dihedral group among free products. This framework extends and strengthens the results of \cite{MS} and \cite{GK}.
In this work, we explore the following question: If two words in a finitely generated free group have identical images as word maps on every finite group, must they be endomorphic to each other? In this regard, we introduce weak profinite rigidity for words, a parallel to profinite rigidity, as defined in \cite{hanany2020some}. We establish that the powers of primitive words in any finitely generated free group $F_n$ are weakly profinitely rigid. Furthermore, if a word in $F_n$ has the same image on every finite group as a test word in $F_n$, then both words induce the same probability measure on every finite group. We also prove that a test word in $F_n$ is weakly profinitely rigid if and only if it is profinitely rigid. As a consequence, we establish that the powers of surface words, i.e., $(x_1^2\ldots x_n^2)^d$ in $F_n$ and $([x_1,x_2]\ldots [x_{2n-1},x_{2n}])^d$ in $F_{2n}$, for $n \geq 1$ and any integer $d$, are weakly profinitely rigid.
In this paper, we study the $C$-width of HNN extension of a group via its proper isomorphic subgroups and amalgamated free product of two groups via their proper isomorphic subgroups with respect to conjugation invariant generating set. We will also establish that infinite one relator group has infinite $C$-width.
A word $w$ in a free group is called {\em chiral} if there exists a group $G$ such that image of word map corresponding to word $w$ is not closed with respect to inverse. Similarly a group $G$ is said to be {\em chiral} if there exists a word $w$ in free group such that $w$ exhibits chirality on the group $G$. Gordeev et al. \cite{gordeev2018geometry} extended the concept of chirality to introduce $\gamma$-chirality in both cases. We show that the notion of $\gamma$-chirality is equivalent to chirality.
A word $w$ in a free group is {\em achiral} if for every group $G,$ $G_w=G_{w^{-1}},$ where $G_w$ is the image of the word map $w$ on $G.$ We will give few classes of examples of achiral words. Cocke and Ho asked whether Engel words are achiral or not. We will prove that it is enough to apply Whitehead's algorithm to check the same.