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Aug 23 
comment 
Torsionfree group that is not of type F but is virtually of type F
I edited to include finite presentation. As YCor says, the BB examples cannot be used for this question and the current conjecture FP+fp=F would apply. 
Aug 23 
revised 
Torsionfree group that is not of type F but is virtually of type F
added 359 characters in body 
Aug 22 
answered  Torsionfree group that is not of type F but is virtually of type F 
Aug 10 
comment 
Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?
Relevant to the last paragraph: an open subgroup of a finitely generated profinite group is finitely generated. This is closely related to: a finite index subgroup of a finitely generated group is finitely generated. 
Aug 10 
revised 
Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?
simplified multiplicative group 
Aug 9 
comment 
vanishing higher cohomology group for property T group?
@HJRW thanks! fixed. 
Aug 9 
revised 
vanishing higher cohomology group for property T group?
left out cocompact + word choice 
Aug 9 
answered  vanishing higher cohomology group for property T group? 
Aug 8 
comment 
Vector spaces without natural bases
The standard gaussian times a basis for polynomials gives several good choices. 
Aug 5 
comment 
BSD leadingterm coefficient in terms of places without distinction
Instead of setting the Tamagawa number of $\mathbb G_m$ to 1, it is probably better to leave it infinite, but declare it a pole with residue 1. Then this formulation covers not just the BSD leading term, but also the order of vanishing. 
Aug 3 
comment 
Detection of stable homotopy by Ktheory spectra
@მამუკა ჯიბლაძე, Mainly I meant "above the dimension" ! ! ! The word "about" probably snuck from something like "is an isomorphism in degrees above a threshold, which is about the dimension." The dimension is the étale dimension. Usually it's stated for fields, where it is the Galois dimension. For a function field, that is equal to the dimension of the original variety (plus the dimension of the ground field). For a general variety, the Zariski dimension gets added in. 
Jul 31 
comment 
When are isotrivial families split by a finite basechange?
Also, since analytic fundamental groups are discrete, this allows the creation of an analytic bundles of abelian varieties over a smooth complete base (say, an elliptic curve) that is isotrivial but not trivial on a finite cover. These is a very different example than the Hodge surface. 
Jul 31 
answered  When are isotrivial families split by a finite basechange? 
Jul 30 
answered  BSD leadingterm coefficient in terms of places without distinction 
Jul 29 
comment 
Detection of stable homotopy by Ktheory spectra
QL is a precise statement about how $K(R)$ is almost $KU$local. Etale $K$theory is $KU$local and QL says that the comparison map is an isomorphism about the dimension. BL adds more detail. (Etale $K$theory is $KU$local because it is the global sections of a sheaf whose stalks are $KU$.) 
Jul 29 
comment 
When are isotrivial families split by a finite basechange?
This is a question where the answer is different in the analytic and algebraic settings. There is an analytic smooth isotrivial family of elliptic curves over $P^1$, the Hodge surface. 
Jul 23 
comment 
BorelSerre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$
The complex structure is natural. The boundary of the locally symmetric space is a subquotient of the boundary of the symmetric space, from which it inherits local properties. As you said, the boundary of 3d hyperbolic space is $\mathbb C\mathbb P^1$; in general, the boundary of hyperbolic space has a unique conformal structure preserved by the group of hyperbolic isometries. 
Jul 23 
comment 
Is the Tate conjecture known for etale covers of products of curves
The key point is that the fundamental group of a product is the product of the fundamental groups. This is only true for complete varieties, but given the context of the Tate conjecture the varieties are complete. 
Jul 23 
comment 
Detection of stable homotopy by Ktheory spectra
Just the image of J. Nothing can detect more than $K(\mathbb Z)$, because everything factors through that. Moreover, everything remains almost $KU$local. 