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Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

2 votes
0 answers
137 views

Image of the pure braid group under the Artin presentation into the automorphism group of th...

As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms. There are a relationship between the mapping class …
qkqh's user avatar
  • 347
2 votes
0 answers
137 views

Term and theories about "relation-free" elements in a group?

For a group $G$, there are two elements a, b which are "relation-free", i.e., there is no nonempty, reduced word $W(X,Y)$ such that $W(a,b)=1$ in $G$. Is there any terminologies or theories for that? …
qkqh's user avatar
  • 347
4 votes
0 answers
208 views

Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the p …
qkqh's user avatar
  • 347
2 votes
1 answer
500 views

Do commutator functor and intersection commute?

For two subgroups $A, B$ in $G$, $[A,A] \cap [B,B] = [A\cap B, A \cap B]$? At least, if $G$ is free, is the left contained in the right?
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  • 347