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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 $r$ satisfyingRelators 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.

Minimality of the conjugacy representative implies that $w$ is freely reduced; otherwise free reduction produces a shorter representative of the same element. It is also a cyclically reduced word. If $w=sv{s}^{-1}$ as a freely reduced word with its first and last letters inverse, the shorter word $v$ represents a conjugate, contradicting minimality. This is the cyclic reduction of a shortest conjugacy representative.

The word is nonempty because the element has order greater than one. Since it is cyclically reduced, no cancellation occurs between successive copies in $w^n$, and $w^n$ is a nonempty freely reduced word. It represents the identity because conjugation preserves the order. Apply the defining property of the Dehn presentation to obtainThe relator belongs to the finite symmetrized set fixed in the definition. No assumption that the group itself is torsion-free is made.

Put $m=|w|$ and $\ell =|r|$. In fact minimality yields the stronger bound $m\le \lfloor \ell /2\rfloor$. Suppose instead thatBecause $|u|>\ell /2$, the first $k$ letters of the matched segment $u$ form a relator segment $v$ of length $k$. They also form a segment of the periodic word $w^n$. Since $k\le m$, that segment is contained in one cyclic permutation $w^{\prime }$ of $w$, starting at the same position in the period. A cyclic permutation represents a conjugate and has length $m$.

Rotate the symmetrized relator to write it as $vs$. Replacing the prefix $v$ of $w^{\prime }$ by $s^{-1}$ preserves the represented element and gives a word of length at mostFree reduction can only shorten further. This contradicts the choice of $w$. ThusThe periodic-segment argument treats matches that cross a boundary between copies of $w$; it would be incorrect simply to assume that the entire long segment $u$ lies in the original written copy of $w$.

Assume first that the finite relator set $R$ is nonempty and putFor every nonidentity finite-order element choose a shortest conjugacy representative. The preceding parts give a word representing its conjugacy class of length at most $M$. The identity class has the empty representative.

There are only finitely many such words because the generating alphabet is finite. If it has $q=2|A|$ formal letters and $N=\lfloor M\rfloor$, a sufficient upper bound for the number of words isDistinct classes cannot require more representatives than there are words; different words may of course represent the same class. ConsequentlyThis is the torsion conjugacy bound for a Dehn presentation. It does not claim that there are only finitely many finite-order elements.

If $R$ is empty, the maximum in the PDF is undefined. Handle this case separately: the group is a free group on the finite alphabet, and a nonempty cyclically reduced word has no freely trivial positive power. Thus there is no nonidentity torsion and only the identity conjugacy class. If the alphabet is empty, the group is trivial and the same conclusion holds.

Use the region labels and curve orientations from part (a). A Dehn presentation of a knot group generator goes from the basepoint above the projection plane through its region and returns through the unbounded region. Equivalently it is dual to the correspondingly labeled $\alpha$ disk of the Heegaard diagram. A crossing relator records the signed intersections of its $\beta$ curve with these disks. Setting $o=1$ givesFor example, the overpassing strand at $A$ equates the meridian of a knot words $d$ and $a{e}^{-1}$; at $B$ it equates $a{d}^{-1}$ and $b{c}^{-1}$. The other two crossings equate $b{a}^{-1}$ with $c{e}^{-1}$ and $c{d}^{-1}$ with $e$.

Choose the region-generator direction so that $e$ is the shown meridian of a knot $\mu$. The Alexander numbering of the regions, starting from zero outside and increasing by one on crossing an oriented strand from right to left, isOne can also obtain this directly by abelianization of the relators: $a=d+e$, $c=d+e$, $b=a-d+c$, and $b-a+e-c=0$. The last two imply $d=e$, and the resulting first homology group is freely generated by $e$. Thus $H_1(X_8;\mathbb{Z})\cong \mathbb{Z}$, with no extra torsion.

Keep all four relators and all five generators, in the orders $(r_A,r_B,r_C,r_D)$ and $(d,e,a,b,c)$. Under abelianization, these generators become $(t,t,t^2,t^3,t^2)$. The Fox calculus rules areFor example, $r_B=a{d}^{-1}c{b}^{-1}$ gives the nonzero Fox derivative entries $-a{d}^{-1}$, $1$, $-a{d}^{-1}c{b}^{-1}$, $a{d}^{-1}$ in columns $d,a,b,c$, respectively. After abelianization these are $-t,1,-1,t$. The full Alexander matrix isThe Fox calculus identity supplies a useful check:Deleting the $e$ column, adjacent to the unbounded region across the marked meridian of a knot, givesThus $\det B$ is a representative of the Alexander polynomial of a knot of the figure-eight knot, with its customary ambiguity of multiplication by $\pm t^n$.

A piece is a nontrivial combinatorial path that occurs in two essentially distinct positions on oriented cell attaching paths. These may be different cells or distinct admissible occurrences on one attaching path; occurrences identified by the defining attaching-path symmetry are not counted twice. In a symmetrized group presentation, it is a common initial segment of distinct relator words. Internal arcs of a reduced disc diagram are pieces, because otherwise the neighboring cells form a cancellable pair.

A finite group presentation has a recursive upper bound on its Dehn function exactly when its word problem for a group is decidable. If the maximal relator length is $\ell$ and a word has length $n$, a diagram of area at most $N$ can be chosen with at most $\ell N+n$ edges. Enumerating all finite labelled planar diagrams within that bound gives a terminating test. Conversely, a word decider identifies the finitely many null words of length at most $n$; exhaustive finite diagram searches find their minimal areas, whose maximum computes the Dehn function.

Small cancellation theory controls overlap between relators or attaching paths of polygonal two-cells. Requiring overlaps to be small compared with cell perimeters gives strong disc diagram structure, shortening algorithms and geometric restrictions on subgroups. The metric small cancellation condition $C^{\prime }(1/6)$ is a central example.

A set of relator words is symmetrized if it includes their inverses and cyclic permutations. Symmetrizing a finite set remains finite, preserves each relator's length and presents the same group. It permits a matching relator segment to be taken anywhere on either orientation of a cell boundary.

In a finite Dehn presentation, every nonidentity finite-order element has a shortest conjugacy representative of length at most half the length of some relator. Indeed a positive power contains a Dehn segment. If the representative had length at least $k=\lfloor |r|/2\rfloor +1$, the first $k$ letters of that segment would fit into a cyclic rotation and could be replaced by fewer letters, contradicting minimality. Bounded relator lengths and a finite alphabet therefore give finitely many conjugacy classes of finite-order elements. With no relators the group is free and torsion-free, so only the identity class remains.

A closed combinatorial path in a 2-complex is null-homotopic exactly when it is the boundary path of a disc diagram over that complex. A finite cellular null-homotopy can be arranged into such a planar diagram; conversely the contractible diagram supplies a null-homotopy. In a group presentation, this is the geometric form of expressing a trivial word as a product of conjugates of relators.