At the moment I'm just reading through this article on the word problem for groups (https://projecteuclid.org/euclid.bams/1183548590), and I'm wondering about a certain snippet.
On page 40 the article describes a word that can encode the state of a turing machine, namely $S_{k_{u}}...S_{k_{1}}q_{i}S_{j_{1}}...S_{j_{v}}$ where $S_{k_{u}}...S_{k_{1}}S_{j_{1}}...S_{j_{v}}$ is the tape expression, $q_{i}$ is the current internal state of the turing machine, and $S_{j_{1}}$ is the currently scanned symbol.
I'm just wondering why this word is necessarily finite. Why can't the input of the turing machine have an infinite amount of non-zero characters?