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This is not strictly an answer to your question, but I hope that results proving that certain nice groups are not IF are also of interest. You point out that free groups and surface groups are IF. Re …

Stallings' folding algorithm (described by Agol) is probably the best way of doing this, but I thought I'd mention an older algorithm which is also useful, due to Whitehead. Let $\{w_k\}$ be a collect …

Many examples can be exhibited using a theorem of Gersten: Theorem (Gersten): Let $G$ be a hyperbolic group of cohomological dimension 2. Every finitely presented subgroup $H$ of $G$ is hyperbolic. Th …

This seems to be true and, like many facts about free groups, can be proved using Stallings' famous folding technique. Here's a sketch of the argument. Think of $F(B)$ as the fundamental group of a b …

The assertion about the number of relations seems to be a theorem of Brunner, proved in the paper Brunner, A. M. Transitivity-systems of certain one-relator groups. Proceedings of the Second Internat …

As @YCor says in comments, there is a finitely generated example due to Ol'shanskii, which is essentially a kind of Tarski monster. However, Ol'shanskii's construction is very complicated. For "nicer" …

This is true, though the proof uses some heavy machinery! Theorem A.1 of Agol--Groves--Manning's appendix to Agol's proof of the Virtual Haken conjecture states: Let $G$ be a hyperbolic group, le …

The case $n=2$ (originally due to Lyndon) admits a very nice geometric argument: one notes that elements $a,b,c$ with $[a,b]=c^2$ lead to a map from the surface $\Sigma_{-1}$ of Euler characteristic - …

You should look up the work on 'residual finiteness growth' (aka 'Farb growth') by Khalid Bou-Rabee and his coauthors.

It's actually easier to prove the stronger statement that there are infinitely many double cosets $H\backslash F/K$. First, note that if $F'$ is a subgroup of finite index in $F$, with $H'$ and $K'$ …

As I said in a comment on your other questions, there is no such word. Indeed, for any $w\in F_n$, there is $u\in F_{n-1}$ so that $w$ survives under the retraction $F_n\to F_{n-1}$ that sends the l …

The answer to both questions is 'no'. This was proved by Greenberg for Fuchsian groups. One outline of the proof is as follows. Any finitely generated subgroup $H$ of a surface group $\Gamma$ is qua …

As Derek Holt says in comments, the answer to your first question is 'yes'. You can argue topologically. There is a rose $R$ corresponding to $X$ with $\pi_1R\cong F$. The subset $A$ defines a conn …

This calculation was performed by Marshall Hall Jr. Let $N(d,n)$ be the number of subgroups of index $d$ in the free group of rank $n$. Then $N(d,n)=d(d!)^{n-1}-\sum_{i=1}^{d-1}((d-i)!)^{n-1}N(i,n)$ …

This is just to flesh out the details of my comment above. I think we can show that $n_k=1$ or $p=1$. After conjugating (and simplifying notation slightly), your equation easily becomes $p^n=[w,b]$ …