A <Dehn presentation> is a finite <group presentation> with the following strict shortening property: every nonempty <freely reduced word> representing the identity contains a consecutive segment $u$ of a <relator> $r$ satisfying
$$
\boxed{|u|>\tfrac12|r|.}
$$
<Relators> are taken as <cyclically reduced words> with a <symmetrized relator set>: their inverse words and cyclic permutations are included. This convention allows the matched segment to start anywhere on either orientation of a <relator>. The symmetrized set remains finite and has the same maximum <relator> length.

If $r=uv$, the relation replaces $u$ by $v^{-1}$, of length $|r|-|u|<|u|$. The <Dehn algorithm> alternates this replacement with <free reduction>. Every step shortens the word; it terminates at the empty word exactly for words representing the identity. The strict inequality matters: a half-perimeter match alone need not shorten anything.

Finiteness is part of the convention used here and is needed for the maximum and counting argument in the last part. With an unrestricted infinite <relator> set, merely imposing the shortening property would not by itself justify that conclusion.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 133 / 4 / a / Solution

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For a <Dehn presentation>, repeatedly freely reduce a word or replace a <relator> segment longer than half its perimeter by its shorter complementary segment. Length decreases strictly. A word represents the identity exactly when the procedure reaches the empty word: any nonempty terminal null word would contradict the defining Dehn property. The finite list of <relators> makes each search effective.

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 Incoming links: Dehn algorithm