Seirios
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 Apr 9 comment Is the Cayley graph of a word-hyperbolic group a CAT(0) metric space? Alternatively, CAT(0) spaces are simply connected (in fact, contractible), so trees are the only CAT(0) graphs. Apr 6 awarded Revival Apr 4 comment Subgroup of free products is torsion-free You're right. In fact, my argument only shows that $\langle g^p,h^q \rangle$ is free for some $p,q \geq 1$. The problem is not as simple as I suspected first. I will delete my answer. Apr 3 answered Subgroup of free products is torsion-free Apr 2 awarded Nice Question Apr 1 awarded Enlightened Apr 1 awarded Nice Answer Mar 30 revised A short exact sequence with a locally finite group edited body Mar 30 comment A short exact sequence with a locally finite group But $a$ and $b$ does not belong to $F$. I use that $[a,b^k] \in N$ for every $k \geq 1$, but a priori the $[a,b^k]$'s does not belong to a finitely-generated subgroup. Mar 29 revised A short exact sequence with a locally finite group added 89 characters in body Mar 29 asked A short exact sequence with a locally finite group Mar 20 answered Infinite groups whose non-trivial subgroups are of finite index Mar 18 comment Prove that a space having the fixed point property must be connected. I essentially gave the answer. Where is your problem? Mar 17 awarded Popular Question Mar 17 answered quasi-geodesics in hyperbolic space Mar 16 comment quasi-geodesics in hyperbolic space Can you precise the point you find uncorrect? Mar 13 awarded Notable Question Mar 7 comment Hyperbolic groups from Dehn functions Thank you for the reference, it sounds interesting. Mar 4 comment Hyperbolic groups from Dehn functions ...We get a new diagram $D'_n$ bounded by the (combinatorial) geodesics $\alpha_n$, $\beta_n$, $[p_n,r_n]$ and $[q_n,s_n]$. Now, we want to say that, if the distances $d(p_n,q_n)$ and $d(r_n,s_n)$ are sufficiently large, then $D'_n$ contains many cells. However, we have to justify that the distance between the geodesics $[p_n,r_n]$ and $[q_n,s_n]$ cannot be too small. Is it reasonnable? Mar 4 comment Hyperbolic groups from Dehn functions Whatever, proving that a linear Dehn function implies the uniform thiness of the bigons seems to be interesting. The argument I have in mind is the following: suppose that there a sequence of bigons $(\alpha_n , \beta_n)$ such that there exists $x_n \in \alpha_n$ and $y_n \in \beta_n$ satisfying $d(x_n,y_n) \to + \infty$; let $D_n$ be a van Kampen diagram whose boundary is precisely this bigon. Now, take some vertices $p_n,q_n \in \alpha_n$ and $r_n,s_n \in \beta_n$, such that $x_n$ is between $p_n$ and $q_n$ along $\alpha_n$ and $y_n$ is between $r_n$ and $s_n$ along $\beta_n$...