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Pekka Nousiainen proved in his PhD thesis "On the Jacobian problem in positive characteristic" at Pennsylvania State University, 1981, a version of the Jacobian Conjecture mod $p$. The results were ne …

It was shown in [Bernhard Herwig and Daniel Lascar, Extending partial automorphisms and the profinite topology on free groups, Trans. Amer. Math. Soc. 352 (2000), 1985-2021] that every finite tourname …

I have seen "(universal) enveloping group of the monoid" used for this construction. Mal'cev has found necessary and sufficient conditions for injectivity of the comparison map. Anatoliy I.Mal’cev, Ü …

The result depends on the approximation properties of $\alpha$. Of course one has to assume $\int_{S^1} f(z)dz=0$. A rotation by $\alpha$ has the effect that the $k$-th Fourier coefficient of $f$ is …

The spectral measures for self-adjoint elements in $\mathbb C F_2$ are very special. In particular, it is known that non of the elements in $\mathbb C F_2$ has a kernel when acting via the left-regula …

If $\langle X,R \rangle$ is a finite presentation of a group $G$, then there exists an exact sequence of $\mathbb ZG$-modules $$0 \to \pi_2(Z) \to \mathbb{Z} G^{\oplus R} \to \mathbb Z G^{\oplus X} \t …

The modified question has a positive answer if $N$ is finitely generated. Consider an extension $1 \to N \to G \to \mathbb Z \to 1$ and take a lift $u \in G$ of the generator of $\mathbb Z$. If $N$ i …

This is not true. The most prominent examples of non-residually finite central extensions of residually finite groups (by $\mathbb Z$) are certain lattices in non-linear Lie groups. See for example …

Dehn, M.; Ueber den Rauminhalt. (German) Math. Ann. 55 (1901), no. 3, 465–478 according to MathSciNet and Springer confirms this here. But on the scanned original provided by the Göttingen Center for …

Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\ …

If $G$ is a finite abelian group, then $\mathbb C[G] = \lbrace f \colon \hat G \to \mathbb C \rbrace$, where $\hat G$ is the Pontrjagin dual of $G$. The isomorphism $g \mapsto g^{-1}$ translates into …

Artin's Conjecture says that any positive integer, which is not a square, is a primitive root modulo infinitely many primes. Christopher Hooley gave in Hooley, Christopher (1967). "On Artin's conjec …

This was proved in the PL-setting in: McMillan, D. R.; Zeeman, E. C. On contractible open manifolds. Proc. Cambridge Philos. Soc. 58 1962 221–224. From MathReviews: "An open manifold is defined to …

The commutant of the action of $M \otimes M^{op}$ on $L^2(M) \otimes_2 L^2(M)$ is $M^{op} \bar \otimes M$ with the obvious action, whereas the commutant of the action on $L^2(M) \otimes_2 L^2(M) \otim …

The answer is yes, this always holds. Note that $$\dim(im(f)) \cdot \|f\|^2 \cdot | {\rm supp}(f)| \geq \tau(f^*f) \cdot |{\rm supp}(f)| \geq |G| \cdot \|f\|^2_1.$$ Here, $\tau \colon \mathbb C[G …