Search ResultsShowing 1-8 of 8
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Semi-equivelar toroidal maps and their vertex covers
Categories: math.COIf the face\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\mbox{-}equivelar. A map is called minimal if the number of vertices is minimal. We know the bounds of number of vertex orbits of semi-equivelar toroidal maps. These bounds are sharp. Datta \cite{BD2020} has proved that every semi-equivelar toroidal map has a vertex-transitive cover. In this article, we prove that if a semi-equivelar map is $k$ orbital then it has a finite index $m$-orbital minimal cover for $m \le k$. We also show the existence and classification of $n$-sheeted covers of semi-equivelar toroidal maps for each $n \in \mathbb{N}$.
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arXiv:2102.09524 (Published 2021-02-18)
The number of configurations in the full shift with a given least period
Comments: 7 pagesFor any group $G$ and set $A$, consider the shift action of $G$ on the full shift $A^G$. A configuration $x \in A^G$ has \emph{least period} $H \leq G$ if the stabiliser of $x$ is precisely $H$. Among other things, the number of such configurations is interesting as it provides an upper bound for the size of the corresponding $\text{Aut}(A^G)$-orbit. In this paper we show that if $G$ is finitely generated and $H$ is of finite index, then the number of configurations in $A^G$ with least period $H$ may be computed using the M\"obius function of the lattice of subgroups of finite index in $G$. Moreover, when $H$ is a normal subgroup, we classify all situations such that the number of $G$-orbits with least period $H$ is at most $10$.
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arXiv:2002.07460 (Published 2020-02-18)
Similarity Isometries of Point Packings
Comments: 13 pages, 6 figuresA linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $\Gamma \subseteq \mathbb{R}^d$ if there exists a positive real number $\beta$ such that $\beta R\Gamma$ is a sublattice of (finite index in) $\Gamma$. The set $\beta R\Gamma$ is referred to as a similar sublattice of $\Gamma$. A (crystallographic) point packing generated by a lattice $\Gamma$ is a union of $\Gamma$ with finitely many shifted copies of $\Gamma$. In this study, the notion of similarity isometries is extended to point packings. We provide a characterization for the similarity isometries of point packings and identify the corresponding similar subpackings. Planar examples will be discussed, namely, the $1 \times 2$ rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, we also consider similarity isometries of point packings about points different from the origin. In particular, similarity isometries of a certain shifted hexagonal packing will be computed and compared with that of the hexagonal packing.
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arXiv:1712.01418 (Published 2017-12-04)
Three-dimensional maps and subgroup growth
Comments: 16 pages, 6 figuresFirstly, we derive a generating series for the number of free subgroups of finite index in $\Delta^+ = \mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z}_2$ by using a connection between free subgroups of $\Delta^+$ and certain three dimensional maps known as pavings, and show that this generating series is non-holonomic. We also provide a non-linear recurrence relation for its coefficients. Secondly, we study the generating series for conjugacy classes of free subgroups of finite index in $\Delta^+$, which correspond to isomorphism classes of pavings. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, and the equivalent types of pavings and their isomorphism classes.
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arXiv:1708.03842 (Published 2017-08-13)
Free subgroups of free products and combinatorial hypermaps
Comments: 25 pages, 3 figures, supplementary material and a SAGE worksheet available at http://sashakolpakov.wordpress.com/list-of-papers/We derive a generating series for the number of free subgroups of finite index in $\Delta^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $\Delta^+$ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the numbers above based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in $\Delta^+$, which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.
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arXiv:1608.08508 (Published 2016-08-30)
The number of ideals of $\mathbb{Z}[x]$ containing $x(x-α)(x-β)$ with given index
Categories: math.COIt is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $\langle B \rangle$ denote the subring of the full matrix ring generated by $B$. Then $\langle B \rangle$ is a free $\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $\langle B \rangle$ with given finite index. Thus, the formal Dirichlet series $\zeta_{\langle B \rangle}(s)=\sum_{n\geq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $\langle B \rangle$ with index $n$. In this article we aim to find an explicit form of $\zeta_{\langle B \rangle}(s)$ when $B$ has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, $\langle B \rangle$ is isomorphic to $\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$ where $m(x)$ is the minimal polynomial of $B$ over $\mathbb{Q}$, and $\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$ is isomorphic to $\mathbb{Z}[x]/m(x+\gamma)\mathbb{Z}[x]$ for each $\gamma\in \mathbb{Z}$. Thus, the problem is reduced to counting the number of ideals of $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x]$ with given finite index where $0,\alpha$ and $\beta$ are distinct integers.
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Dynamical properties of profinite actions
Comments: Corrections made based on the referee's commentsSubjects: 37C85We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($\tau$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($\tau$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.
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K^F-invariants in irreducible representations of G^F, when G=GL_n
Comments: Revised version, 38 pages; same content, improved expositionJournal: J. Algebra 261 (2003), no. 1, 102--144Keywords: irreducible representations, general result, explicit formulas, frobenius map, finite indexTags: journal articleUsing a general result of Lusztig, we give explicit formulas for the dimensions of K^F-invariants in irreducible representations of G^F, when G=GL_n, F:G->G is a Frobenius map, and K is an F-stable subgroup of finite index in G^theta for some involution theta:G->G commuting with F. The proofs use some combinatorial facts about characters of symmetric groups.