scholarly journals A Generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups

2007 ◽  
Vol 124 (1) ◽  
pp. 37-67
Author(s):  
Lewis Bowen
Keyword(s):  
Conjugacy Classes ◽  
Prime Geodesic Theorem ◽  
Free Subgroups

scholarly journals Three-dimensional maps and subgroup growth

2021 ◽  
Author(s):  
Rémi Bottinelli ◽  
Laura Ciobanu ◽  
Alexander Kolpakov
Keyword(s):  
Finite Index ◽  
Three Dimensional ◽  
Conjugacy Classes ◽  
Computational Experiments ◽  
Subgroup Growth ◽  
Isomorphism Classes ◽  
Generating Series ◽  
Free Subgroups ◽  
Finite Index Subgroups

AbstractIn this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on n darts, thus solving an analogue of Tutte’s problem in dimension three. The generating series we derive also counts free subgroups of index n in $$\Delta ^+ = {\mathbb {Z}}_2*{\mathbb {Z}}_2*{\mathbb {Z}}_2$$ Δ + = Z 2 ∗ Z 2 ∗ Z 2 via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $$\Delta ^+$$ Δ + on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in $$\Delta ^+$$ Δ + . Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $$n\le 16$$ n ≤ 16 darts.

scholarly journals ASYMPTOTIC FORMULAS FOR CLASS NUMBER SUMS OF INDEFINITE BINARY QUADRATIC FORMS ON ARITHMETIC PROGRESSIONS

2012 ◽  
Vol 09 (01) ◽  
pp. 27-51 ◽  
Author(s):  
YASUFUMI HASHIMOTO
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Modular Group ◽  
Conjugacy Classes ◽  
Arithmetic Progressions ◽  
Asymptotic Formulas ◽  
Binary Quadratic Forms ◽  
Congruence Subgroups ◽  
Arithmetic Properties ◽  
Prime Geodesic Theorem

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants on arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the remainder terms are sharper.

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

scholarly journals Groups with boundedly finite conjugacy classes of commutators

2018 ◽  
Vol 69 (3) ◽  
pp. 1047-1051 ◽  
Author(s):  
Gláucia Dierings ◽  
Pavel Shumyatsky
Keyword(s):  
Conjugacy Classes ◽  
Finite Conjugacy

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

scholarly journals A classification of spherical conjugacy classes

2016 ◽  
Vol 285 (1) ◽  
pp. 63-91
Author(s):  
Mauro Costantini
Keyword(s):  
Conjugacy Classes

A NOTE ON THE NUMBER OF ZEROS IN THE COLUMNS OF THE CHARACTER TABLE OF A GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250150 ◽  
Author(s):  
JINSHAN ZHANG ◽  
ZHENCAI SHEN ◽  
SHULIN WU
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Table ◽  
Number Of Zeros

The finite groups in which every irreducible character vanishes on at most three conjugacy classes were characterized [J. Group Theory13 (2010) 799–819]. Dually, we investigate the finite groups whose columns contain a small number of zeros in the character table.

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