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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 …

This is probably a trivial observation, but maybe useful to track the difficult part of the problem. If instead of condition three you require: (strongly exhaustive) There exists some constant $C>0 …

Modulo a (possibly severe) measurability issue, the only groups which admit an invariant submeasure are the amenable groups. Let $G$ be a non-amenable group and $\mu$ be a $G$-invariant submeasure o …

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 …

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 …

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 …

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 …

This is more a remark, since I do not directly answer the question. The statement in the question is true for all ideals $I,J$ (without the condition $I+J=R$) if and only if all ideals a idempotent, i …

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 …

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 …

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 …

One has $Hom({\mathbb C}[F_n],{\mathbb C}) = ({\mathbb C}^{\times})^n$ with the obvious topology. (Here, $Hom$ denotes the space of $\mathbb C$-linear homomorphisms.) This of course uses a little bit …

I think I figured it out myself. What was meant is that for every finite subset $S$ of $A_{\delta}$ one has $$| \lbrace (n,m) \in S \times S \mid n-m \in A_{\delta^2/2} \rbrace | \geq \frac{\delta^2 | …

Let $G \subset SL_n(\mathbb C)$ be a Zariski closed subsemigroup. The map $$\alpha(x,y) := (x,xy)$$ defines an injective self-map of $G \times G$ (see as algebraic varieties over $\mathbb C$). By the …

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 …