ترغب بنشر مسار تعليمي؟ اضغط هنا

Meta-nilpotent quotients of mapping-torus groups and two topological invariants of quadratic forms

55   0   0.0 ( 0 )
 نشر من قبل Takefumi Nosaka
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English
 تأليف Takefumi Nosaka




اسأل ChatGPT حول البحث

We determine the center of a meta-nilpotent quotient of a mapping-torus group. As a corollary, we introduce two invariants, which are quadratic forms, of knots and of mapping classes.

قيم البحث

اقرأ أيضاً

260 - Takefumi Nosaka 2021
We develop nilpotently $p$-localization of knot groups in terms of the (symplectic) automorphism groups of free nilpotent groups. We show that any map from the set of conjugacy classes of the outer automorphism groups yields a knot invariant. We also investigate the automorphism groups and compute the resulting knot invariants.
We explain a correspondence between some invariants in the dynamics of color exchange in a 2d coloring problem, which are polynomials of winding numbers, and linking numbers in 3d. One invariant is visualized as linking of lines on a special surface with Arf-Kervaire invariant one, and is interpreted as resulting from an obstruction to transform the surface into its chiral image with special continuous deformations. We also consider additional constraints on the dynamics and see how the surface is modified.
Let $n, k geq 3$. In this paper, we analyse the quotient group $B_n/Gamma_k(P_n)$ of the Artin braid group $B_n$ by the subgroup $Gamma_k(P_n)$ belonging to the lower central series of the Artin pure braid group $P_n$. We prove that it is an almost-c rystallographic group. We then focus more specifically on the case $k=3$. If $n geq 5$, and if $tau in N$ is such that $gcd(tau, 6) = 1$, we show that $B_n/Gamma_3 (P_n)$ possesses torsion $tau$ if and only if $S_n$ does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order $tau$ in $B_n/Gamma_3 (P_n)$ with those of elements of order $tau$ in the symmetric group $S_n$. We also exhibit a presentation for the almost-crystallographic group $B_n/Gamma_3 (P_n)$. Finally, we obtain some $4$-dimensional almost-Bieberbach subgroups of $B_3/Gamma_3 (P_3)$, we explain how to obtain almost-Bieberbach subgroups of $B_4/Gamma_3(P_4)$ and $B_3/Gamma_4(P_3)$, and we exhibit explicit elements of order $5$ in $B_5/Gamma_3 (P_5)$.
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of th e paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.
We prove the statement in the title and exhibit examples of quotients of arbitrary nilpotency class. This answers a question by D. F. Holt.
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