# The dimension of a torsion-free $p$-adic analytic group generated by two generators

$$\DeclareMathOperator\GL{GL}$$Let $$G$$ be a $$2$$-generator pro-$$p$$-group of finite rank, i.e. it is isomorphic to a closed subgroup of $$\GL_d(\mathbb{Z}_p)$$ for some integer $$d$$. Assume that $$G$$ is torsion-free. Recall that the dimension $$\dim(G)$$ of $$G$$ as a $$p$$-adic analytic group can be described as $$d(U)$$ where $$U$$ is any uniform open pro-$$p$$-subgroup of $$G$$ and $$d(U)$$ denotes the minimal cardinality of a topological generating set for $$U$$.

Question: Is $$\dim(G)=2$$?

• No: for $p \geq 5$, the group $G := \begin{pmatrix}1 + p\mathbb{Z}_p & \mathbb{Z}_p \\ p\mathbb{Z}_p & 1 + p\mathbb{Z}_p\end{pmatrix} \cap \textrm{SL}_2(\mathbb{Z}_p)$ is torsion-free and generated by two elements, but has $\dim(G) = 3$.
– krl
Commented Dec 2, 2022 at 1:47

A pro-$$p$$ group $$G$$ has $$\textit{lower rank}$$ $$r$$ if $$r$$ is minimal such that every open subgroup of $$G$$ contains an open subgroup generated by at most $$r$$ elements. Lubotzky and Mann showed that the lower rank of a $$p$$-adic analytic pro-$$p$$ group is the number of generators of its associated Lie algebra. On the other hand, its dimension is the dimension of the Lie algebra. Thus, for instance, every open subgroup of $$SL_d(\mathbb{Z}_p)$$ (including torsion free ones) has lower rank $$2$$ and dimension $$d^2-1$$.