Geometric group theory

Let $F_2$ be the free group on $a, b$ with generating set $S = \{a^{\pm1}, b^{\pm1}\}$. How many elements does the ball $B_S(3)$ of radius $3$ contain (including the identity)?

Which of these pairs of finitely generated groups are quasi-isometric? (Choose all that apply.)

The group $\mathbb{Z}$ acts on $\mathbb{R}^2$ by $n \cdot (x, y) = (x + n, y)$. Yet $\mathbb{Z}$ is not quasi-isometric to $\mathbb{R}^2$. Which hypothesis of the Švarc–Milnor lemma fails?

In the ping-pong lemma, $X_1, X_2$ are disjoint and non-empty, $a^n(X_2) \subseteq X_1$ and $b^n(X_1) \subseteq X_2$ for $n \ne 0$. For $w = a^{2} b^{-5} a^{-1}$, which inclusion does the proof establish, and why does it show $w \ne 1$?

The integer Heisenberg group $H_3(\mathbb{Z})$ has lower central series $G_1 = H_3(\mathbb{Z}) \supset G_2 = Z \supset G_3 = 1$, where $Z \cong \mathbb{Z}$ is the centre, and $G_1 / G_2 \cong \mathbb{Z}^2$. By the Bass–Guivarc'h formula, what is the degree $d$ of polynomial growth, $|B(r)| \sim r^d$?

Let sanovA sanovB : SL(2, ℤ) be Sanov's matrices. Which Lean statement says that they generate a free group of rank 2?

Which of these groups are amenable? (Choose all that apply.)

Which statement about the Banach–Tarski paradox is correct?

Write ball_size(generators, multiply, identity, r) returning $|B(r)|$, the number of group elements that are products of at most $r$ generators, by breadth-first search. The generators are assumed closed under inverses. Test it on $\mathbb{Z}$ with generating set $\{\pm 2, \pm 3\}$, where $|B(1)| = 5$ and $|B(2)| = 13$.