Iain Aitchison, Lawrence Reeves
We associate to each finite presentation of a group G a compact CW-complex that is a 3-manifold in the complement of a point, and whose fundamental group is isomorphic to G. We use this complex to define a notion of genus for G and give examples, and also define a notion of `closed group'. A group has genus 0 if and only if it is the fundamental group of a compact orientable 3-manifold.
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We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite completions.
In this article we calculate the profinite genus of the fundamental group of a $n$-dimensional compact flat manifold $X$ with holonomy group of prime order. As consequence, we prove that if $n\leq 21$, then $X$ is determined among all $n$-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of profinite completion of the fundamental group of $X$ in terms of the representation genus of its holonomy group.
We show that if the action of a classical group $G$ on a set $Ω$ of $1$-spaces of its natural module is of genus at most two, then $|Ω| \leq 10,000$.
For a finite group $G$ denote by $γ(L(G))$ the genus of the subgroup graph of $G.$ We prove that $γ(L(G))$ tends to infinity as either the rank of $G$ or the number of prime divisors of $|G|$ tends to infinity.
In this article we study the extent to which an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order.
In 1962 M.J.Wicks gave a list of forms for commutators in both free groups and free products. Since then similar lists have been constructed for elements of higher genus. A. Vdovina has described a method for the construction of forms for elements of any genus in free products. We give a similar result for the construction of such forms in any hyperbolic group H and from this we obtain a full list of forms for commutators in H.
The commuting graph of a non-abelian group is a simple graph in which the vertices are the non-central elements of the group, and two distinct vertices are adjacent if and only if they commute. In this paper, we classify (up to isomorphism) all finite non-abelian groups whose commuting graphs are acyclic, planar or toroidal. We also derive explicit formulas for the genus of the commuting graphs of some well-known class of finite non-abelian groups, and show that, every collection of finite non-abelian groups whose commuting graphs have the same genus is finite.
We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we classify all families of affine genus zero systems; that is all families of coverings of the Riemann sphere by itself such that the monodromy group is a primitive affine permutation group.
In this paper we establish lower and upper bounds for the cardinality of the profinite genus of the fundamental group $π_{1}(M_A)\cong (\mathbb{Z} \times \mathbb{Z})\rtimes_{A}\mathbb{Z}$ of a torus bundle $M_{A}$ in terms of the number of ideal classes of the order $\mathbb{Z}[λ]$, where $λ$ is an eigenvalue of the matrix $A$ in $\mathrm{GL}_{2}(\mathbb{Z})$.
Motivated by the need for efficient isomorphism tests for finite groups, we present a polynomial-time method for deciding isomorphism within a class of groups that is well-suited to studying local properties of general finite groups. We also report on the performance of an implementation of the algorithm in the computer algebra system {\sc magma}.
The profinite genus of the groups $\mathbb{Z}^n\rtimes C_{p^2}$
本文给出了形如$\mathbb{Z}^n \rtimes C_{p^2}$的群的 profinite 流形公式，完成对$\mathbb{Z}^n \rtimes G$半直积群的流形大小计算，其中G是有限p-群且具有有限整数表示类型。
Exponential equations in free groups were studied initially by Lyndon and Schutzenberger and then by Comerford and Edmunds. Comerford and Edmunds showed that the problem of determining whether or not the class of quadratic exponential equations have solution is decidable, in finitely generated free groups. In this paper we show that for finite systems of quadratic exponential equations decidability passes, under certain hypotheses, from the factor groups to free products and one-relator products.
We show that if $\cal S$ is a compact Riemann surface of genus $g = p+1$, where $p$ is prime, with a group of automorphisms $G$ such that $|G|\geqλ(g-1)$ for some real number $λ>6$, then for all sufficiently large $p$ (depending on $λ$), $\cal S$ and $G$ lie in one of six infinite sequences of examples. In particular, if $λ=8$ then this holds for all $p\geq 17$ and we obtain the largest groups of automorphisms of Riemann surfaces of genenera $g=p+1$.
If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there exists a monomorphism of $G$ into $H$ which induces isomorphisms between the corresponding quotients of their lower central series. We first consider residually nilpotent groups and find sufficient conditions on the monomorphism so that $H$ is para-$G.$ We then prove that for certain polycyclic groups, if $H$ is para-$G$, then $G$ and $H$ have the same Hirsch l…
The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. In this paper we complete the calculation of the strong symmetric genus for each finite Coxeter group excluding the group E8.
The strong symmetric genus of a group is the smallest genus of a surface on which the group acts faithfully as a group of orientation preserving automorphisms. In this paper we announce and prove the strong symmetric genus for the hyperoctahedral groups. This parameter is already known for the alternating and symmetric groups as well as the sporadic finite simple groups.
The strong symmetric genus of a finite group is the minimum genus of a compact Riemann surface on which the group acts as a group of automorphisms preserving orientation. A characterization of the infinite number of groups with strong symmetric genus zero and one is well-known and the problem is finite for each strong symmetric genus greater than or equal to two. May and Zimmerman have published papers detailing the classification of all groups with strong symmetric genus two through four. Using the computer algebra system GAP, we extend these classifications to all groups of strong symmetric …
We derive double coset formulae for the genus and extended genus of a finitely generated nilpotent group G, using the notions of bounded and bounded above automorphisms of $\prod G_S$, which are defined relative to a fixed fracture square for G.
By definition, an $ω$-residually free tower is positive-genus if all surfaces used in its construction are of positive genus. We prove that every limit group is virtually a subgroup of a positive-genus $ω$-residually free tower. By combining this with results of Gaboriau, we prove that elementarily free groups are measure equivalent to free groups.
We compute the number of orbit types for simply connected simple algebraic groups over algebraically closed fields as well as for compact simply connected simple Lie groups. We also compute the number of orbit types for the adjoint action of these groups on their Lie algebras. We also prove that the genus number of a connected reductive algebraic group coincides with the genus number of its semisimple part.
Purely pseudo-Anosov subgroups of the genus two handlebody group
本文证明了亏格2柄体群中有限生成的纯伪Anosov子群具有凸余紧性。
We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups to genus-one hyperbolic 2-bridge knot groups.
The genus spectrum of a finite group $G$ is the set of all $g$ such that $G$ acts faithfully on a compact Riemann surface of genus $g$. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the abelian $p$-groups. Motivated by the work of Talu for odd primes $p$, we develop a general combinatorial machinery, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of abelian $p$-groups. We have a particular view towards how to generally find the reduced minimum genus in this class of group…
The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a {\it universal mapping class group of genus zero}, denoted $\B$. The latter is a nontrivial extension of the Thompson group $V$ (acting on the Cantor set) by an inductive limit of pure mapping class groups of all genus zero surfaces. We prove that $\B$ is a finitely presented group, and give an explicit presentation of it.
We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.
Every group is the automorphism group of a graph with arbitrarily large genus
本文证明任意抽象群都可实现为某类图的自同构群，且这些图的亏格可随参数无界增长。
We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in topological dynamics.
We classify, up to conjugacy, the finite (constant) subgroups G of adjoint absolutely simple algebraic groups of type $A_1$ over an arbitrary field $k$ of characteristic not 2.
Stable Subgroups of the Genus Two Handlebody Group
本文证明亏格2柄体群的有限生成子群稳定当且仅当到圆盘图的轨道映射为拟等距嵌入，通过证明其为分层双曲群并利用相关刻画完成，同时给出应用并指出更高亏格类似不成立。
Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper we obtain finite presentation for the mapping class group of the closed non-orientable surface of genus 4 from its action on the so called ordered complex of curves.
The "relating" entwines three problems: 1. Davenport's Problem, describing pairs of polynomials over Q whose ranges on Z/p are the same for almost all p. 2. Showing that the monodromy groups of rational function maps over the complexes are limited to a finite set of groups, outside of groups close to alternating groups (example, symmetric groups) with special representations, and dihedral and cyclic groups. 3. Relating the genus 0 modular curves to the character group of the Monster simple group, so-called Monstrous Moonshine. http://www.math.uci.edu/~mfried/pathlist-cov/thomp-genus0.html has …
For any hyperbolic genus one 2-bridge knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ lies in some range which depends on the knot.
We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus $g$ handlebody group into the genus $g$ mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection of the genus $g$ handlebody group into a genus $h>g$ mapping class group which is not conjugate into a handlebody group.
Isometry groups of infinite-genus hyperbolic surfaces
本文对无平面端的无限亏格二维流形的常曲率黎曼度量等距群给出近乎完全分类，证明具自相似端空间的流形可实现所有可数群为等距群，并推导相关群的性质阻碍与映射类群代数刚性结果。
The power graph $Γ_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have (non)orientable genus two.
Signatures of alternating group actions with non-zero quotient genus
本文研究了交替群作用于紧黎曼面时商亏格大于0的特征，证明当n>6时所有可能的特征都是实际存在，n=5或6时仅[1;2]或[1;3]失效，并证明非阿贝尔单群的所有商亏格大于1的特征都是实际存在。
We first provide an overview of several results dealing with the genus of a division algebra and highlight the role of ramification in its analysis. We then give a survey of recent developments on the genus problem for simple algebraic groups and its connections to the analysis of groups with good reduction.
Monodromy groups of indecomposable coverings of bounded genus
本文分类了亏格为 $g$ 的复曲线不可分解覆盖的分歧类型和单演群，并证明了 Guralnick 及 Guralnick 和 Shareshian 的相关猜想。
Homological representations of low genus mapping class groups
本文研究了有限群作用于曲面时，其中心化子在同调上的表示是否具有算术像，并给出了亏格至多3时的完整答案。
We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $ρ(g-1)$ for some integer $ρ\ge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for $ρ>6$, and of the first and third authors for $ρ=3, 4, 5$ and $6$. As a corollary we classify the orientably regular hypermaps (including maps) of genus $p+1$, together with the non-orientable regular hypermaps of characteristic $-p$, with automorphism group of order divisible by the prime $p$; thi…
Finite group actions on genus two $SL(2, \mathbb{C})$-character variety and applications to SCFTs
本文研究了有限群作用下亏格为二的SL(2,C)特征种集的固定点集，并探讨其在4d N=2 SCFTs中的应用。
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