Failure of the $L^1$ pointwise and maximal ergodic theorems for the free group

Let $F_2$ denote the free group on two generators $a,b$. For any measure-preserving system $(X, {\mathcal X}, μ, (T_g)_{g \in F_2})$ on a finite measure space $X = (X,{\mathcal X},μ)$, any $f \in L^1(X)$, and any $n \geq 1$, define the averaging operators ${\mathcal A}_n f(x) := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f( T_g^{-1} x ),$ where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f \in L^1(X)$ such that the sequence ${\mathcal A}_n f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maximal ergodic theorems do not hold in $L^1$ for actions of $F_2$. This is despite the results of Nevo-Stein and Bufetov, who establish pointwise and maximal ergodic theorems in $L^p$ for $p>1$ and for $L \log L$ respectively, as well as an estimate of Naor and the author establishing a weak-type $(1,1)$ maximal inequality for the action on $\ell^1(F_2)$. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.

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