MarriageTheoremのこと

_ arXiv:math 10月18日分まで、IACR ePrint 2012/605まで確認済み

_ 気になった論文１：On trivial words in finitely presented groups, M. Elder, A. Rechnitzer, E. J. Janse van Rensburg, T. Wong, http://jp.arxiv.org/abs/1210.3425

We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of such groups by finding the cogrowth series for Baumslag-Solitar groups $BS(N,N) = \langle a,b | a^N b = b a^N \rangle$ and prove that their cogrowths are algebraic numbers. We have been unable to find the cogrowth series for other Baumslag-Solitar groups, but we have found recurrences that yield the first few terms of the cogrowth series exponentially faster than is possible by naive methods. Finally we apply our numerical method to several presentations of Thompson's group $F$ and our results give strong indication that the group is not amenable.

_ 気になった論文２：Group structures of elliptic curves over finite fields, Vorrapan Chandee, Chantal David, Dimitris Koukoulopoulos, Ethan Smith, http://jp.arxiv.org/abs/1210.3880

It is well-known that if $E$ is an elliptic curve over the finite field $\F_p$, then $E(\F_p)\simeq\Z/m\Z\times\Z/mk\Z$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\le M$ and $k\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\Z/m\Z\times\Z/mk\Z$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\le (\log M)^{2-\epsilon}$, then a density zero proportion of the groups in question actually arise as the group of points on some elliptic curve over some prime finite field. On the other hand, if $K\ge (\log M)^{2+\epsilon}$, they conjectured that a density one proportion of the groups in question arise as the group of points on some elliptic curve over some prime finite field. We prove that the first part of their conjecture holds in the full range $K\le (\log M)^{2-\epsilon}$, and we prove that the second part of their conjecture holds in the limited range $K\ge M^{4+\epsilon}$. In the wider range $K\ge M^2$, we show that a positive density of the groups in question actually occur.

_ 気になった論文３：On conjugacy separability of some Coxeter groups and parabolic-preserving automorphisms, Pierre-Emmanuel Caprace, Ashot Minasyan, http://jp.arxiv.org/abs/1210.4328

We prove that even Coxeter groups, whose Coxeter diagrams contain no (4,4,2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.

_ 気になった論文４：Randomness, pseudorandomness and models of arithmetic, Pavel Pudlak, http://jp.arxiv.org/abs/1210.4692

Pseudorandmness plays an important role in number theory, complexity theory and cryptography. Our aim is to use models of arithmetic to explain pseudorandomness by randomness. To this end we construct a set of models $\cal M$, a common element $\iota$ of these models and a probability distribution on $\cal M$, such that for every pseudorandom sequence $s$, the probability that $s(\iota)=1$ holds true in a random model from $\cal M$ is equal to 1/2.