Geometric group theory

Geometric group theory studies finitely generated groups through their actions on metric spaces and through the large-scale geometry of spaces such as their Cayley graphs.

Choose a finite symmetric generating alphabet $\Sigma$ and one word $\sigma (g)$ representing each group element, with $\sigma (1)$ the empty word. The associated paths in the Cayley graph start at the identity and traverse one edge per unit time, remaining at their endpoint after completion. A synchronous combing has a constant $K$ such that paths to adjacent endpoints stay within $K$ at equal times. An asynchronous version permits increasing reparametrizations. The formal language need not be a regular language; regular synchronous representative languages define automatic groups.

A finitely generated group is synchronously combable if it admits a combing of a group with the synchronous fellow-traveller property. Without a regular-language assumption, this is more general than an automatic group. A bounded-rung strip argument gives a finite group presentation, and finite-state row shortening gives a recursive exponential filling bound and a soluble word problem for a group.

Attach relators for every identity word of length at most $2K+2$, where $K$ is an integer fellow-traveller constant. For any null loop, connect its consecutive combing paths at integer heights by rungs of length at most $K$. Each resulting quadrilateral has perimeter at most $2K+2$ and thus is a relator. The bottom row is collapsed at the identity and the top row is the original loop, so these finitely many relators fill every null word and present the group.

For a null loop of length $n$, a horizontal row of its combing diagram is encoded by an ordered $n$-tuple of rung words of length at most $K$. If there are $D$ such words, there are at most $D^n$ row types. Between two identical labelled rows one can delete the entire intervening annular band and glue the equal boundary paths. Repeating this leaves at most $D^n$ layers of $n$ bounded-perimeter cells. Thus a synchronous combable group has a recursive exponential isoperimetric bound. This does not require the combing language to be regular or computable.

Two paths satisfy the synchronous fellow-traveller property with constant $K$ if $d(p(t),q(t))\le K$ for all equal times, with completed paths held at their endpoints. Checking integer times suffices up to increasing $K$ by at most two for unit-speed edge paths. A synchronous combing of a group requires a uniform constant for all adjacent endpoints.

An automatic group admits an automatic structure for a group: a regular language of representatives and synchronous group multiplier automata recognizing right multiplication by each generator and equality of representatives. Representatives need not be unique. This finite-state description yields effective algorithms for the word problem for a group.

An automatic structure for a group consists of a finite generating alphabet $A$, a surjective representative regular language $L\subseteq A^{\ast }$, and finite-state automata accepting the padded convolutions of words $(u,v)$ in $L$ satisfying $\overline{u}x=\overline{v}$ for each $x\in A\cup \{\epsilon \}$. Symmetry of $A$ and uniqueness of representatives are additional conditions, not part of this definition.

A group multiplier automaton accepts exactly the padded convolutions of words $u\otimes v$ with $u,v\in L$ and $\overline{u}x=\overline{v}$. The case $x=\epsilon$ recognizes equality in the group, rather than equality of literal words over an alphabet.

For a fixed finite generating alphabet and integer $K$, use the elements of the word-metric ball $B_K(1)$ as states, together with a reject state and padding-status flags. Starting at the identity, a padded letter pair $(u,v)$ changes the state to $u^{-1}dv$ if it stays in the ball. A padding letter acts as the identity. Final state $a$ recognizes endpoint difference $a$ among pairs whose prefixes remain within $K$. On a regular synchronous combing, intersecting with the two word acceptors gives the full group multiplier automaton. It does not recognize unrestricted endpoint relations for all input words.

Fix the first tape of a group multiplier automaton and use dynamic programming over automaton states and output-padding status, retaining predecessor pointers. The pumping of a group multiplier automaton bounds the search by $|u|+N$ layers, each of fixed size. This constructs a representative of $\overline{u}x$ in time $O(|u|+1)$ without assuming uniqueness of representatives.

In an accepted padded convolution of words, a loop after one tape ends can be deleted or repeated while keeping that entire tape fixed. The altered other tape still represents the same required neighbouring group element. A shortest representative has a suffix no longer than the fixed number of automaton states; an accepted longer suffix gives infinitely many representatives of the same element.

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 CW subcomplex has no missing shells if a cell with a shell-length boundary arc contained in the CW subcomplex is itself contained in the CW subcomplex. For $C^{\prime }(1/6)$ it suffices that every excluded cell have a complementary boundary arc of length at least half its perimeter with no edge in the CW subcomplex: an arc lying in the CW subcomplex then has length at most half and cannot be a shell's exterior arc.

For the metric small cancellation condition $C^{\prime }(1/6)$, every nonempty freely reduced null-homotopic combinatorial word contains a segment of a cell boundary longer than half its perimeter. Such a segment can be replaced by the shorter complementary segment. The diagram explanation uses shells with at most three internal pieces: their internal length is less than half the perimeter. Degenerate diagrams require the disc diagram spur alternative.

A reduced $C^{\prime }(1/6)$ disc diagram is a single vertex, a single closed cell, a disc diagram ladder, or contains at least three exposed disc diagram spurs and/or disc diagram shells with at most three interior pieces. This is the strong diagram form of the Greendlinger lemma. It follows from the combinatorial curvature inequality after suppressing valence-two vertices: if there are not three positively curved exposed features, all remaining blocks must occur in a chain. In a nontrivial multi-block ladder its two ends are exposed features. Marking two boundary corners and excluding shells/spurs in the side interiors therefore forces the ladder alternative. This is the width-one case of McCammond and Wise, Theorem 9.4; reduced $C^{\prime }(1/6)$ diagrams satisfy its arc-reduced $C(6)$–$T(3)$ hypotheses.

The condition $C^{\prime }(\lambda )$ requires that every piece in small cancellation theory occurring in a cell boundary have length strictly less than $\lambda$ times that cell's perimeter. Length counts edges. For $\lambda =1/6$, an inner path made of at most three pieces has length strictly less than half the perimeter, so the complementary outer path is strictly longer. The attaching paths are combinatorial immersions and the piece convention is part of the chosen presentation or polygonal complex.

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.

An isometric group action on a metric space is cobounded if some orbit is within a fixed distance $R$ of every point. Equivalently, translates of one bounded ball cover the space. Coboundedness does not require finite stabilizers, a proper space, or a finitely generated group. Every action on a point is cobounded.

A CAT(0) space is a geodesic metric space whose geodesic triangles are no thicker than their Euclidean comparison triangles.

A Hadamard space is a complete CAT(0) space. Equivalently, every pair has a midpoint $m$ satisfying the midpoint inequality $d(z,m{)}^2\le \frac{1}{2}d(z,x{)}^2+\frac{1}{2}d(z,y{)}^2-\frac{1}{4}d(x,y{)}^2$ for every $z$. This inequality makes the midpoint unique. Iterating midpoints and using completeness constructs the unique geodesic joining the pair. The squared-distance inequality extends to every point of that geodesic by dyadic approximation and implies Euclidean triangle comparison and geodesic convexity of closed balls.

For the constant-speed geodesic $\gamma$ from $x$ to $y$ in a Hadamard space, the displayed inequality follows by repeatedly applying the midpoint inequality and passing from dyadic parameters to all $t\in [0,1]$. Applying it on two sides of a triangle gives $d(x_s,y_t{)}^2\le s^2{A}^2+t^2{B}^2-st(A^2+B^2-C^2)$, where $A=d(a,x)$, $B=d(a,y)$ and $C=d(x,y)$. The right side is the squared comparison distance by the Euclidean cosine rule.

A cube complex is formed by gluing Euclidean cubes along isometric faces so that intersecting cubes meet along common faces.

The link of a vertex $v$ records the local directions from $v$: its vertices are incident edge germs, and a collection spans a simplex exactly when the corresponding edge germs lie in a common cube.

A cube complex is nonpositively curved when every vertex link is a flag simplicial complex. Equivalently, its universal cover with the piecewise Euclidean metric is a CAT(0) space.

A CAT(0) cube complex is a simply connected nonpositively curved cube complex.

Two edges of a cube complex are parallel when they are opposite sides of a square, with transitive closure understood. A hyperplane is assembled from the midcubes dual to one parallelism class of edges.

The carrier $N(H)$ of a hyperplane $H$ is the union of the closed cubes that meet $H$.

A nonpositively curved cube complex is special when every hyperplane is embedded and two-sided, no hyperplane self-osculates, and no two hyperplanes interosculate. Equivalently, it admits a cubical local isometry to a Salvetti complex.

A hyperplane self-intersects when two adjacent edges of one square are dual to that same hyperplane.

A hyperplane is one-sided when its carrier is a twisted interval bundle rather than a product with an interval.

A two-sided hyperplane self-osculates when two distinct, consistently oriented edges dual to it have the same initial vertex but do not form the corner of a square.

Two hyperplanes interosculate when they cross in one square and also have dual edges meeting at a vertex without spanning a square elsewhere.

The local isometry from a connected special cube complex to a Salvetti complex induces an injection of fundamental groups. Hence its fundamental group is a subgroup of a Right-angled Artin group.

A wallspace $(S,\mathcal{W})$ is a set $S$ with walls $W=\{W^{+},W^{-}\}$, each a partition into two halfspaces, such that only finitely many walls separate any two points of $S$.

Such wall partitions encode separation data used to construct a dual cube complex.

A wall is an unordered pair of complementary halfspaces $\{W^{+},W^{-}\}$.

A halfspace is one of the two complementary parts determined by a wall.

Two walls cross when all four intersections of one halfspace from each wall are nonempty. A collection of pairwise crossing walls is also called transverse.

A transverse collection of walls is a collection in which every two distinct walls cross.

The wall metric is

The vertices of the dual cube complex are consistent choices of one halfspace of every wall that differ from a principal choice on only finitely many walls. Two vertices are joined when their choices differ on one wall, and higher cubes fill the resulting hypercubes. The result is a CAT(0) cube complex.

A point $x\in S$ determines a principal vertex by choosing, for every wall, the halfspace containing $x$. The combinatorial distance between the principal vertices of $x$ and $y$ is $d_{\mathcal{W}}(x,y)$.

A cubulation of a group is a metrically proper action by cubical automorphisms on a CAT(0) cube complex. It is cocompact when the action is also a cocompact group action.

An isometric group action on a metric space is metrically proper when, for some and hence every point $x$, the set $\{g:d(x,gx)\le R\}$ is finite for every finite $R$.

If $G$ acts on $(S,\mathcal{W})$ and $x\in S$, then its action on the dual cube complex is metrically proper wheneveras $g$ leaves every finite subset of $G$.

The action of $G$ on the dual cube complex is cocompact exactly when there are finitely many $G$-orbits of finite transverse wall collections. Equivalently, there is a uniform bound on their sizes and, for every size, only finitely many orbits.

For a graph $\Gamma$, the right-angled Artin group is

The Salvetti complex $S_{\Gamma }$ has one vertex, one oriented loop for each vertex of $\Gamma$, and an $n$-torus cubulated by one $n$-cube for every $n$-clique of $\Gamma$. It is nonpositively curved and ${\pi }_1{S}_{\Gamma }\cong A_{\Gamma }$.

The unique vertex of $S_{\Gamma }$ has two link vertices $v^{+},v^{-}$ for each $v\in V(\Gamma )$. Signed vertices $v_1^{{ϵ}_1},\dots ,v_k^{{ϵ}_k}$ span a simplex exactly when the distinct underlying vertices $v_1,\dots ,v_k$ form a clique of $\Gamma$.

The hyperplane of $S_{\Gamma }$ dual to the loop labelled $v$ is naturally the Salvetti complex of the induced subgraph on the neighbours of $v$.

Every right-angled Artin group is a residually finite group. One route is its faithful integral linear representation, followed by reduction modulo suitable primes.

The free group $F(S)$ on a set $S$ consists of reduced words in letters $s^{\pm 1}$, with multiplication followed by cancellation. Every map from $S$ to a group extends uniquely to a homomorphism from $F(S)$.

An endomorphism of $F_n$ has image generated by at most $n$ elements. If this image has finite index $d$, the Nielsen–Schreier formula gives rank $1+d(n-1)$. Its abelianization requires at least that many generators. For $n\ge 2$, comparison with $n$ forces $d=1$.

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In $F(a,b)$, any set of distinct conjugates $b^{-i}a{b}^i$ freely generates its subgroup. Combine adjacent powers of the same conjugate in a reduced word. After expansion, distinct successive indices leave nonzero intervening powers of $b$, and the intervening nonzero powers of $a$ cannot cancel in the free product $⟨a⟩\ast ⟨b⟩$. The same argument works with $a,b$ interchanged and for finite or countably infinite sets of indices.

If $a,b$ freely generate, then $a^{i}b{a}^{-i}$ for any distinct integer indices freely generate their subgroup. Group a reduced word in these new generators into blocks with nonzero exponents and different successive indices. Expansion produces alternating nonzero powers of $b$ and interior powers $a^{i_{j+1}-i_j}$, so it cannot freely reduce to the empty word. Consequently every finite-rank free group embeds in a rank-two free group; free rank is not monotone under subgroup inclusion.

Suppose a group homomorphism $\pi :G\to Q$ is injective on a subgroup freely generated by $x_1,\dots ,x_n$. If $y_i$ have $\pi (y_i)=\pi (x_i)$, then the $y_i$ freely generate their subgroup: a relation among them projects to a relation among the free basis images, so the corresponding abstract reduced word must be empty. In particular, this applies to $y_i=x_{i}g$ when the normal closure of $g$ intersects $⟨x_1,\dots ,x_n⟩$ trivially.

A subset $X$ is a free basis of a group $G$ when the natural map from the free group on $X$ to $G$ is an isomorphism. Equivalently, $X$ generates $G$ and no nonempty freely reduced word in $X\cup X^{-1}$ equals the identity. A generating set is not automatically a free basis.

A word in generators and their formal inverses is freely reduced if no consecutive letters are mutual inverses. Deleting such inverse pairs gives the unique reduced representative in a free group. In a general group presentation, free reduction preserves the represented group element but may not produce a shortest representative.

Free reduction removes adjacent inverse-letter pairs. A stack algorithm scans the letters, canceling the top letter when the next is its inverse. The resulting reduced word is unique: deleting an inverse pair before scanning does not change the final stack, so every cancellation sequence gives the same result. In any group presentation these deletions preserve the represented element, and in a free group the reduced word is the identity exactly when it is empty.

A freely reduced word is cyclically reduced if its first and last letters are not inverse, with the empty word also allowed. Every cyclic permutation of a nonempty cyclically reduced word is freely reduced, represents a conjugate, and has the same length. Positive powers are freely reduced because no cancellation occurs at copy boundaries.

A reduced word contains no adjacent generator and inverse-generator pair. Cancel such pairs to obtain the unique reduced representative in a free group. One proof constructs the free group itself from reduced words, with multiplication given by concatenation followed by cancellation; its action by these operations on words distinguishes distinct reduced representatives.

For every function $f:S\to G$ from a set into a group, there is a unique group homomorphism $\bar{f}:F(S)\to G$ whose restriction to $S$ is $f$.

Suppose nontrivial groups $G_1,G_2$ act on a set and there are disjoint nonempty subsets $X_1,X_2$ such that every nonidentity element of $G_1$ maps $X_2$ into $X_1$, while every nonidentity element of $G_2$ maps $X_1$ into $X_2$. Assume at least one factor has at least three elements. Then the subgroup generated by $G_1$ and $G_2$ is their free product. The cardinality hypothesis excludes the counterexample in which $G_1=G_2=C_2$ acts by swapping $X_1$ and $X_2$. Versions allowing both factors to have order two need an additional separation hypothesis.

Every finitely generated free group is a residually finite group. For $F_2$, the ping-pong lemma embeds it into ${\mathrm{SL}}_2(\mathbb{Z})$, and reduction modulo a prime in an integral matrix group separates each nonidentity element in a finite quotient.

A group presentation $⟨S\mid R⟩$ describes the quotient of the free group $F(S)$ by the normal closure of the relators $R$. It is finite when both $S$ and $R$ are finite.

A recursive presentation has an effectively enumerable alphabet of generators and a recursively enumerable set of relators. Usually the alphabet is finite; when it is countably infinite this is specified explicitly. Enumerating finite products of conjugates of relators and their inverses enumerates all null words. Conversely the enumerable null words can themselves serve as the relators. Recursive presentability does not imply a soluble word problem for a group.

A finitely generated group embeds in a finitely presented group if and only if it has a recursive presentation of a group. The necessity follows by enumerating proofs that substituted generator words are trivial in the larger finite group presentation. The sufficiency is the embedding theorem; it also applies to effectively countably generated recursive presentations after an effective two-generator embedding. This is the theorem of Subgroups of finitely presented groups.

For a group presentation $\mathcal{P}=⟨X\mid R⟩$ with finitely many generators and a prime number $p$, defineThis uses the unshifted convention. Identity relators have zero weight, and a divergent sum gives value $-\infty$. Some literature subtracts one instead; inequalities must be shifted accordingly. Taking roots in the free group matters: roots appearing only after passing to the quotient do not change the relator weight. The p-rank of a group bounds this quantity above, and the index-p rewriting bound for p-deficiency gives the infinitude criterion p-deficiency at least one implies infinitude.

Enumerate all nonidentity words $w_i$ in the rank-two free group and impose relations $w_i^{p^{i+2}}=1$. The resulting two-generated group hasIt is infinite by p-deficiency at least one implies infinitude. Each element has order a power of $p$, so it is a torsion group. The infinitely many relators are essential to this particular construction; finite generation does not imply a finite group presentation.

If a group presentation has unshifted p-deficiency at least one, the p-rank of a group is at least one and gives a normal subgroup of index $p$. The index-p rewriting bound for p-deficiency gives this subgroup another presentation with p-deficiency at least one. Iterating produces subgroups of index $p^j$ for every $j$, so the group is infinite. This criterion applies even with infinitely many relators, provided the weighted sum in the definition converges.

If $H$ is a normal subgroup of index $p$ in a group presented by $\mathcal{P}$, the Reidemeister–Schreier theorem produces a presentation $\mathcal{Q}$ withThe preimage of $H$ in the free group has rank $1+p(d-1)$. A relator $w^{p^k}$ produces at most $p$ conjugate relators of weight $p^{-k}$ if its root lies in that preimage. If the root lies outside, $k\ge 1$, and the coset conjugates are redundant up to conjugation; one relator of weight at most $p^{-(k-1)}$ suffices. Thus the total weighted relator cost grows by at most $p$. Convergent infinite sums are handled term by term.

For a finitely generated group, let $G^p$ denote the subgroup generated by all $p$th powers. Its p-rank isThis is the dimension of the largest quotient that is an elementary abelian group of exponent $p$. In a group presentation with $d$ generators and $s$ relators that are not $p$th powers, reduction of the exponent sums modulo $p$ gives at most $s$ linear constraints. Hence $d_p(G)\ge d-s\ge {\mathrm{def}}_p(\mathcal{P})$. In particular a presentation of p-deficiency at least one yields a surjection onto $C_p$.

For a fixed finitely generated group with specified generators, its word problem asks for an algorithm deciding whether an input word represents the identity. For a finitely presented group, positive answers can be enumerated from the relators, but there need not be an algorithm deciding both positive and negative answers. An effective reduction from a nonrecursive halting set proves unsolvability.

An amalgamated free product of finite cyclic groups has an effective normal form theorem for an amalgamated free product: multiplication, subgroup membership and subgroup identifications are finite table operations. Replacing an amalgamated syllable by its image in the neighbouring factor and combining adjacent syllables from one factor strictly decreases syllable count. A remaining alternating product outside the amalgamated subgroup is nonidentity. This supplies a terminating word problem for a group algorithm.

For $S\subseteq {\mathbb{N}}_{>0}$, form the Right-angled Artin group on generators $b_i$ indexed by $i\in \mathbb{Z}$, imposing $[b_i,b_j]=1$ precisely when $|i-j|\in S$. Translation $b_i\mapsto b_{i+1}$ is an automorphism. Its semidirect product with an infinite cyclic group generated by $t$ is countable and generated by $b_0,t$. If $n\notin S$, killing all $b_i$ except $b_0,b_n$ retracts the graph group onto the rank-two free group, so their commutator is nonidentity. This proves the displayed equivalence. For non-recursively-enumerable $S$, enumerable identities of finitely generated subgroups rules out an embedding of this two-generated group in any finitely presented group.

If a finitely generated group embeds in a finitely presented group, its null words in any fixed finite generating set form a recursively enumerable set. Choose words for the generator images in the larger group. Enumerate finite products of conjugates of its defining relators and their inverses, freely reduce, and compare with substituted words. Dovetailing over input words enumerates exactly the subgroup's identities. Choosing the finite image words is nonuniform, but their existence suffices for existence of an enumeration algorithm.

A finitely generated group $G$ has a soluble word problem for a group if and only if there are a simple group $S$ and a finitely presented group $P$ with embeddings $G\hookrightarrow S\hookrightarrow P$. The theorem does not require $S$ itself to be finitely presented. In particular, a finitely generated subgroup of a finitely presented simple group has soluble word problem. The primary statement is Theorem I of An algebraic characterization of groups with soluble word problem.

Let $S$ be a nontrivial simple group inside a finitely presented group $P$, and fix $a\ne 1$ in $S$. For an input word $w$ in $S$, enumerate products of conjugates in $P$ of $w^{\pm 1}$ and proofs in $P$ that such a product equals $a$. This search terminates exactly when $w\ne 1$: the normal closure of a nontrivial element of $S$ is all of $S$, and its conjugators are also elements of $P$, while a null word has only trivial conjugates. Run this search alongside the enumerable proofs that $w=1$. Exactly one search terminates, solving the word problem for a group on any specified finitely generated subgroup of $S$. No effective generating set for the possibly infinitely generated $S$ is needed.

A finitely generated group with soluble word problem for a group embeds in a countably generated simple group with a recursive presentation of a group. Start with $P_0=G\ast ⟨a⟩$, retaining decidable membership in $⟨a⟩$. At stage $n$, add a free infinite cyclic generator $z_n$. For every nonidentity base element $g$, conjugate $[g,z_n]$ to $a$ and conjugate $g{z}_n$ to $a$; also conjugate $z_n$ to $a$, using distinct stable letters. All associated subgroups are infinite cyclic, with decidable membership by effective cyclic membership in a free product. Britton's lemma preserves embeddings, decidable word problem and decidable membership in the inherited $⟨a⟩$.

In the union, any nontrivial normal subgroup contains some nonidentity $g$ and hence $a$, by the commutator-conjugation relation. The other relations force $z_n$ and then every old $g$ into the normal closure of $a$. Every stable letter becomes an old element at a later stage, so the normal closure of $a$ is the whole union. The union is nontrivial and simple. Effective word enumeration at each stage makes all relators recursively enumerable. A two-generator HNN embedding followed by the Higman embedding theorem embeds this simple envelope in a finitely presented group.

A finite group presentation has finitely many generators and finitely many relators. It defines the quotient of the free group on $X$ by the normal closure of $R$. Free products of finitely presented groups and HNN extensions along explicitly finitely generated associated subgroups have finite presentations obtained by adjoining finitely many generators and relations.

A finitely presented group is a group admitting a finite group presentation. Finite presentability concerns the existence of such a presentation, not the decidability of its word problem for a group.

Suppose $G=⟨Y\mid R(Y)⟩$ is a finite group presentation, and $X$ is any finite generating set. Write $y_j=u_j(X)$ and $x_i=v_i(Y)$. A finite presentation on $X$ is obtained using the relators $R(u(X))$ and $x_i=v_i(u(X))$. The substitutions in both directions give inverse homomorphisms, proving that finite presentability does not depend on the finite generators chosen.

// Target: geometry-and-topology.bigb

The Higman group has the finite group presentation $Q=⟨a_1,a_2,a_3,a_4\mid a_i{a}_{i+1}{a}_i^{-1}=a_{i+1}^2(i\mathrm{mod}4)⟩$. Its subgroup $⟨a_1,a_3⟩$ is a rank-two free group, and the normal closure of each $a_i$ is all of $Q$. It has no nontrivial finite quotients of a group: a least prime divisor among the four generator orders in a finite image contradicts the multiplicative-order constraints imposed by conjugation to squares. These properties give finite-presentation constructions embedding arbitrary finitely presented groups into groups with no nontrivial finite quotients.

The two three-generator components on $\{a,b,c\}$ and $\{c,d,a\}$ are iterated HNN extensions of infinite cyclic groups. The subgroup on $\{a,c\}$ is free of rank two in each component by Britton's lemma. Amalgamating over these free subgroups gives the displayed infinite finitely presented group. Its cyclic squaring relations forbid every nontrivial finite quotient by the least-prime multiplicative-order argument.

A group presentation with a cyclic list of generators $a_i$ and relations $a_i{a}_{i+1}{a}_i^{-1}=a_{i+1}^2$ has no nontrivial finite quotient of a group. In a finite image, all generator orders are odd. Choose the least prime $p$ dividing any nontrivial generator order and let $x$ be the predecessor of a generator $y$ whose order is divisible by $p$. Conjugation by $x$ acts as squaring on $⟨y⟩$, so the multiplicative order of $2$ modulo $p$ divides the order of $x$. That multiplicative order is greater than one and divides $p-1$, giving a prime divisor smaller than $p$ in the order of $x$, a contradiction. This argument applies to any cycle length; it asserts absence of finite quotients, not infinitude of the presented group.

An isomorphism-invariant property of finitely presented groups is Markov when some finitely presented group has it and some finitely presented group cannot embed in any finitely presented group having it. The second witness is an obstruction to embedding, rather than merely a group failing the property. The Adian–Rabin theorem makes every such property algorithmically undecidable from arbitrary finite presentations.

No Markov property of finitely presented groups is decidable by an algorithm taking an arbitrary finite group presentation as input. The proof reduces the word problem for a group to property recognition using an effective construction which collapses when an input word is trivial and embeds the input group when it is nontrivial.

In the effective group form used for recognition problems, a finite group presentation of $K$ and a word $w$ yield, by an algorithm, a finite presentation of a group $K_w$. If $w=1$ in $K$, then $K_w$ is trivial; if $w\ne 1$ in $K$, then the specified generators embed $K$ in $K_w$. One can additionally arrange that the normal closure of the image of $w$ is all of $K_w$. The finite presentation can be written without first deciding which case holds. This is the collapse-versus-embedding construction behind the Adian–Rabin theorem.

A Tietze transformation changes a group presentation without changing the presented group, by adding or removing a consequence of the relators, or by adding or eliminating a generator together with a defining equation expressing it in the others. For example a generator $t$ defined by $t=xs{x}^{-1}$ can be eliminated by substitution in every remaining relator.

A finite group presentation with cyclically reduced symmetrized relators is a Dehn presentation if every nonempty freely reduced null word contains a relator segment longer than half that relator. Replacing the segment by the inverse complementary segment strictly shortens the word. Finiteness of both the alphabet and the relator set is part of this definition; merely permitting all null words as an infinite relator set would make the shortening property vacuous as a finiteness criterion.

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.

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.

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.

A map from the generators $S$ of $⟨S\mid R⟩$ to a group $G$ extends to a homomorphism exactly when every relator in $R$ evaluates to the identity. The extension is then unique.

A relator is a word in the generators of a free group that is imposed as the identity in a group presentation.

A free presentation of a group is a quotient $G\cong F/R$ in which $F$ is a free group and $R$ is the normal closure of the chosen relators.

A finite group presentation is balanced when it has the same number of generators and relators.

For a finite group presentation, the relator exponent-sum matrix records in row $i$ the total exponent of each generator in relator $r_i$. It presents the abelianization of the group. For a balanced presentation of a perfect group, it is a unimodular matrix.

For a free presentation $G\cong F/R$, the relation module is the abelianization $R/[R,R]$ equipped with the $G$-action induced by conjugation in $F$.

If $I_F$ is the augmentation ideal of $\mathbb{Z}F$, a free presentation $G\cong F/R$ gives an exact sequenceThe middle module is free over $\mathbb{Z}G$ on the free generators of $F$.

Fox calculus assigns noncommutative derivatives $\partial /\partial x_i$ to words in a free group. The product rule and the identityturn relators into module maps and give the presentation relation sequence.