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Let $G_0$ be a finitely generated group, and suppose there are groups $G_i$ and $K_i$ as in the following short exact sequences $1\to K_i\to G_{i+1}\to G_i\to 1$ with $K_i$ free and nonabelian (you …

This is a follow-up question to Is there a non-free group $G$ whose subgroups are all freely decomposable? In the answer to that question, Cornulier gives the following example (due to Kurosh) of a gr …

The answer is $K=2k$, which is an exercise. Here is a sketch: Let $f\colon \langle x_i\rangle\to G$ be a surjective homomorphism, and $h\colon \langle x_i\rangle*\langle y\rangle\to G$ the map defined …

It is well known that in a given $\delta$-hyperbolic group there are only finitely many conjugacy classes of finite subgroups. This is clearly false for relatively hyperbolic groups since we have no c …

Let $X_w$ be the presentation complex of a one-relator group $\langle x_1,\dotsc,x_n\mid w\rangle$, with $w$ cyclically reduced, i.e., $X_w=R\cup_w D$, with $R$ the rose with $n$ petals labeled $x_1,\ …

Stallings proves the following (remarkable theorem!) using the lower central series. (Homology and central series of groups, p171.) Suppose $G$ is a group, $K$ a field, with $H_2(G,K)=0$. If $x_1,\do …

I don't think so. This boils down to finding a retraction from the group $\langle a,b,c,d,q,r,t\rangle/\langle\langle tQabRcdT=QbaRdc\rangle\rangle$ to the subgroup $\langle a,b,c,d\rangle$. The White …