6,883 reputation
11729
bio website normalesup.org/~cornulier
location
age
visits member for 3 years, 6 months
seen 2 hours ago

1d
comment Which groups are LERF?
Hint: LERF implies residually finite.
1d
comment Marshall Hall's theorem for surface groups
See the references here: arxiv.org/abs/1204.5135. It includes the reference to Scott original paper, and an erratum.
1d
comment Marshall Hall's theorem for surface groups
Yes, surface group are LERF, it's Peter Scott's theorem.
Oct
22
comment Which finite p-groups occur as commutators of finite p-groups?
@Dietrich: the link addresses the question whether a NLA is derived of some lie algebra (necessary solvable, but not nilpotent a priori). The analogous question here would rather be which NLA are derived subalgebras of nilpotent Lie algebras.
Oct
19
comment Actions of amenable groups on graphs with uncountably many ends
That's (finally) the question. At the moment, I only can find examples with countably (infinitely) many ends.
Oct
18
comment Actions of amenable groups on graphs with uncountably many ends
Because you can artificially make the graph amenable by taking the pointed union with a bi-infinite line, adding an extra-generator $q$. Since then $[G,q]$ is contained in the set of finitely supporting permutations, you deduce that the new acting group $\langle G,q\rangle$ is amenable (if $G$ is amenable). Hence if you have an amenable group with a Schreier graph with uncountably many ends, you also get an amenable group with an amenable Schreier graph with uncountably many ends.
Oct
18
comment Actions of amenable groups on graphs with uncountably many ends
So amenability of both the group and the Schreier graph are part of the assumptions? this sounds not very natural to me.
Oct
18
comment Actions of amenable groups on graphs with uncountably many ends
How do you see that the Schreier graphs of an amenable group are amenable?
Oct
18
comment Actions of amenable groups on graphs with uncountably many ends
No, it seems you make a confusion. The action of $G$ on $G/H$ endowed with the Schreier graph metric is not by isometries in general.
Oct
18
comment Actions of amenable groups on graphs with uncountably many ends
Your first sentence is confusing to me: if $G$ is a group with a finite generating set and $H$ a subgroup, there is an action of $G$ on $G/H$, a ("Schreier") graph structure on $G/H$, but the action of $G$ on $G/H$ does not preserve the graph structure except in a few rare exceptions. Here are a two distinct interpretations of the question: 1) does there exist a f.g. amenable group with a Schreier graph with uncountably many ends? 2) does there exist a transitive action of a f.g. amenable group on a graph with infinitely many ends?... Could you clarify?
Oct
17
comment On a problem of Berkovich
what's $|\cdot|_p$?
Oct
16
comment Is the regularity of finitely generated rings decidable?
I can see how to index maximal ideals: enumerate all finite fields and homomorphisms into these finite fields and compute generators for the kernels. This can be done algorithmically. But how do you detect when the localization is not regular?
Oct
16
comment Is the regularity of finitely generated rings decidable?
Just to clarify, I guess that the input is numbers $k,m$ and and $m$ polynomials $P_1,\dots,P_m$ in $\mathbf{Z}[X_1,\dots,X_k]$ and you wish to get YES or NO according to whether the ring $R=\mathbf{Z}[X_1,\dots,X_k]/(P_1,\dots,P_m)$ is regular. Are you suggesting that you at least have an algorithm that answers NO if $R$ is not regular? (which means the problem is at least semidecidable)
Oct
13
comment pro-Lie-groups and the exponential map
What is the meaning for a Lie algebra to be abstractly isomorphic to a Lie group? These are not the same structures.
Oct
13
comment Gromov-Hausdorff convergence for non-compact metric spaces
Anyway the same "joke" can hold in a connected space (e.g. use the union of the line $Im(z)=1$ and the segments $[n,n+i]$ for $n\in\mathbf{Z}$ in the complex plane, instead of $\mathbf{Z}$).
Oct
12
comment Milnor-Wolf result on growth of solvable groups
Osin actually improved Chou's result, see arxiv.org/abs/math/0404075, and indeed had to go into the technicalities. He proves uniform exponential growth. Actually there are a few other improvements that are known for non-vn f.g. solvable groups, but not known for elementary amenable groups, such as: - the existence of a QI-embedded free subsemigroup on 2 generators - exponential conjugacy growth.
Oct
11
comment Milnor-Wolf result on growth of solvable groups
But the original proof has the advantage to generalize to elementary amenable groups (Chou, 1980).
Oct
11
comment Milnor-Wolf result on growth of solvable groups
There's one proof of 2 using the fact, due to Groves (1978), that a non-vn f.g. solvable group has a homomorphism into $K^*\ltimes K$ with non-vn image for some non-discrete locally compact field $K$, and using this we can play ping-pong on the tree/hyperbolic 2/3-space, without distinction between the polycyclic and non-polycyclic case.
Oct
11
comment Milnor-Wolf result on growth of solvable groups
I could check Milnor's paper but not Wolf's. Milnor (J. Diff Geom. 2, p447-449, 1968) checks that if $G$ is f.g. solvable but not polycyclic then it has exponential growth. His proof actually shows it contains a free subsemigroup (see the reasoning by contradiction in his Lemma 1), although he doesn't say it.
Oct
11
comment Milnor-Wolf result on growth of solvable groups
Would you say something about what the original proof is? the only proof I have in mind consists in proving 1) if $G$ is f.g. virtually nilpotent, then it has polynomial growth 2) if $G$ is f.g. solvable and not virtually nilpotent, then it has a free subsemigroup. There are probably several proofs of (2).