bio  website  normalesup.org/~cornulier 

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10h

revised 
Quotients of finitely generated nilpotent groups
for clarification, edited to remove the statement that Woess claimed the stronger statement 
17h

comment 
On sentences true in all finite groups (revisited)
About (2): The sentence $\exists x\forall y_1,\dots,y_k:w(x,y_1,\dots,y_k)=1$ holds in every group iff it holds in a free group of countable rank, iff $w(1,y_1,\dots,y_k)=1$ as abstract word. I'm not sure if the same conclusion if the above sentence is only supposed to hold in every finite group. 
18h

comment 
The possibility of a symmetric difference in a torsionfree group
My guess is that constructing such groups would require a small cancelation argument and some computer assistance in order to check that these (graphical?) small cancelation conditions are fulfulled (by prescribing $B$ to be of some given even large enough cardinal). 
22h

comment 
Quotient group of an amalgam
The 1st question has an obvious answer: if $f:G\to H$ is surjective and $H=A_1\ast_{A_2} A_3$, then $G=B_1\ast_{B_2} B_3$, where $B_i=f^{1}(A_i)$. This sounds unrelated to the second question, which sounds "too broad". 
22h

comment 
Quotients of finitely generated nilpotent groups
The edited question is not of research level and should be erased here and answered on MathSE. The original question sounds natural and interesting if turned into a question instead of a reference request. 
1d

revised 
Group with finite outer automorphism group and large center
Added infinitely generated example, and fixed out>outer in title, added 1 tag 
1d

comment 
Quotients of finitely generated nilpotent groups
here's the MathSE link: math.stackexchange.com/questions/1301400/… 
1d

comment 
Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semisimple Lie group?
sorry for taking so much time, but I wanted to write a full argument, not working only for large $n$. 
1d

answered  Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semisimple Lie group? 
2d

comment 
Free actions of nonamenable groups
rather of $\mathcal{L}^\infty(X)$, the space of bounded measurable functions. 
2d

comment 
A family of posets
sounds different: the 4element poset $b>a<c<d$ has the above property but is not skeletal according to Ex 9.6 in your link 
2d

comment 
The possibility of a symmetric difference in a torsionfree group
It's not possible when the group is biorderable (e.g., abelian or even nilpotent, or free). Indeed, if $x_1B=x_2B\Delta x_3B$ then since $B$ is nonempty, we have $x_1,x_2,x_3$ pairwise distinct, and since this condition is symmetric in $x_1,x_2,x_3$, we can suppose $x_1<x_2<x_3$. Since $B$ is finite, it has a maximal element $b$, so that $x_3b$ belongs to neither $x_2B$ nor $x_1B$, contradicting the condition. 
May 26 
comment 
classification of $p$groups
I can believe this (for class2 nilpotent groups in the complex case it already gets really complicated, maybe the threshold is just a little higher; actually I don't know if the passage from complex to arbitrary fields, say of char. $>11$, is known even in dimension 7.) 
May 26 
comment 
classification of $p$groups
@DerekHolt: it's one point of view but I'm not 100% convinced it's the only one. For instance, complex nilpotent Lie algebras of dimension 8 are classified in the case of nilpotency length 2 (and there are finitely many) while the classification is much more complicated for nilpotency length 7 and 6 (that is, coclass 1 and 2). This should be reflected in the classification of groups of order $p^8$ and exponent $p$ when $p\ge 11$, since their classification can be restated in terms of Lie algebras over $Z/pZ$. 
May 26 
comment 
classification of $p$groups
2: only for a few small values of $n$ (I think the classification of groups of order $p^n$ then tends to be "stable" when $n$ is fixed in the sense that it has a uniform description for $p\ge p_n$ for some $p_n$). 
May 26 
comment 
Nontrivial finite group with trivial cohomology in prescribed degree
A variant: given $i$, does there exist a finite group $G$ such that $H^j(G)=0$ for every $j$ with $0<j\le i$? 
May 26 
comment 
Subgroups of the group $G_2 \times G_2$
A product of groups $H_1\times H_2$ contains a simple group $S$ iff either either $H_1$ or $H_2$ contains $S$. Idem for Lie algebras. 
May 26 
comment 
Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$
If you replace the target set $\mathbf{N}$ with a 2element set, the components of the resulting graph are all isomorphic and called "hypercubes". When the target set is $\mathbf{N}$ or $\mathbf{Z}$, it is also natural to put an edge only when $fg$ is $\pm$ the Dirac function at some point, so that the resulting distance is the $\ell^1$distance. 
May 25 
comment 
Hyperbolic manifold of dim 3 with finite volume.
this was my guess, but you could reformulate your first sentence as something "By work of Agol, Wise and others, the Virtual Haken theorem provides a classification of closed hyperbolic 3manifolds (up to finite covering, and modulo a suitable classification of pseudoanosov elements in the mapping class groups)." 
May 25 
comment 
Hyperbolic manifold of dim 3 with finite volume.
By the way I don't understand if you're speaking of 3folds or hyperbolic 3folds, so I may misunderstand your question. The geometrization theorem is a decomposition theorem for 3folds, which for a geometrizable 3fold such as a hyperbolic 3fold is a tautology. 