ترغب بنشر مسار تعليمي؟ اضغط هنا

Hyperbolic reflection groups associated to the quadratic forms $-3x_0^2 + x_1^2 + ... + x_n^2$

53   0   0.0 ( 0 )
 نشر من قبل John Mcleod
 تاريخ النشر 2010
  مجال البحث
والبحث باللغة English
 تأليف John Mcleod




اسأل ChatGPT حول البحث

We determine the maximal hyperbolic reflection groups associated to the quadratic forms $-3x_0^2 + x_1^2 + ... + x_n^2$, $n ge 2$, and present the Coxeter schemes of their fundamental polyhedra. These groups exist in dimensions up to 13, and a proof is given that in higher dimensions these quadratic forms are not reflective.


قيم البحث

اقرأ أيضاً

44 - Tushar Kanta Naik 2018
Let $P(G)$ denotes the set of sizes of fibers of non-trivial commutators of the commutator word map. Here, we prove that $|P(G)|=1$, for any finite group $G$ of nilpotency class $3$ with exactlly two conjugacy class sizes. We also show that for given $ngeq 1$, there exists a finite group $G$ of nilpotency class $2$ with exactlly two conjugacy class sizes such that $|P(G)|=n$.
55 - Ashvin Swaminathan 2020
Let $n geq 3$ be an integer. In this paper, we study the average behavior of the $2$-torsion in class groups of rings cut out by integral binary $n$-ic forms having any fixed odd leading coefficient. Specifically, we compute upper bounds on the avera ge size of the $2$-torsion in class groups of rings and fields arising from such binary forms. Conditional on a uniformity estimate, we further prove that each of these upper bounds is in fact an equality. Our theorems extend recent work of Bhargava-Hanke-Shankar in the cubic case and of Siad in the monic case to binary forms of any degree with any fixed odd leading coefficient. When $n$ is odd, we find that fixing the leading coefficient increases the average $2$-torsion in the class group, relative to the prediction of Cohen-Lenstra-Martinet-Malle. When $n$ is even, such predictions are yet to be formulated; together with Siads results in the monic case, our theorems are the first of their kind to describe the average behavior of the $p$-torsion in class groups of degree-$n$ rings where $p mid n > 2$. To prove these theorems, we first answer a question of Ellenberg by parametrizing square roots of the class of the inverse different of a ring cut out by a binary form in terms of the integral orbits of a certain coregular representation. This parametrization has a range of interesting applications, from studying $2$-parts of class groups to studying $2$-Selmer groups of hyperelliptic Jacobians.
An abstract group $G$ is called totally 2-closed if $H = H^{(2),Omega}$ for any set $Omega$ with $Gcong Hleqtextrm{Sym}_Omega$, where $H^{(2),Omega}$ is the largest subgroup of symmetric group of $Omega$ whose orbits on $OmegatimesOmega$ are the same orbits of $H$. In this paper, we prove that the Fitting subgroup of a totally 2-closed group is a totally 2-closed group. We also conjecture that a finite group $G$ is totally 2-closed if and only if it is cyclic or a direct product of a cyclic group of odd order with a generalized quaternion group. We prove the conjecture in the soluble case, and reduce the general case to groups $G$ of shape $Zcdot X$, with $Z = Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian
We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group $A$ to be (virtually) cocompactly cubulated, in terms of the defining graph of $A$.
In this paper, we study a group in which every 2-maximal subgroup is a Hall subgroup.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا