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Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups

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 Added by Takefumi Nosaka
 Publication date 2021
  fields
and research's language is English




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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.



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54 - Takefumi Nosaka 2019
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.
We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $mathrm{GL}(n,mathbb{R})$ action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature $s$ under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with $s eq0$. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with $s eq0$ on a nilpotent Lie group. We show that nilpotent Lie groups of dimension $leq 6$ do not admit such a metric, and a similar result holds in dimension $7$ with the extra assumption that the Lie algebra is nice.
If $G$ is a free product of finite groups, let $Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Guti{e}rrez-Krsti{c} [M. Guti{e}rrez and S. Krsti{c}, {em Normal forms for the group of basis-conjugating automorphisms of a free group}, International Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley-Krsti{c} [W. Bogley and S. Krsti{c}, {em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $underline{E} Sigma Aut_1(G)$-space for these groups.
We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension $n$ up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for $nleq9$. On every nilpotent Lie algebra of dimension $leq 7$, we determine the number of inequivalent nice bases, which can be $0$, $1$, or $2$. We show that any nilpotent Lie algebra of dimension $n$ has at most countably many inequivalent nice bases.
Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of $G$ has finite index in the center of $G$ we show that the growth is precisely $n^b$, where $b$ is the product of the nilpotency class and dimension of $G$. In the general case, we give a method for finding an upper bound of the form $n^b$ where $b$ is a natural number determined by what we call a terraced filtration of $G$. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.
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