Editing Word problem for groups (section)

==Examples==
{{more citations needed section|date=December 2023|reason=Citations should be given for individual examples.}}
The following groups have a solvable word problem:
*[[Automatic group]]s, including:
**[[Finite group]]s
**[[Negatively curved group|Negatively curved (aka. hyperbolic) group]]s
**[[Euclidean group]]s
**[[Coxeter group]]s
**[[Braid group]]s
**[[Geometrically finite group]]s
*Finitely generated [[free group]]s
*Finitely generated [[free abelian group]]s
*[[Polycyclic group]]s
*Finitely generated recursively [[Absolute presentation of a group|absolutely presented group]]s,<ref>{{citation |first=H. |last=Simmons |title=The word problem for absolute presentations |journal=J. London Math. Soc. |volume=s2-6 |issue=2 |pages=275–280 |date=1973 |doi=10.1112/jlms/s2-6.2.275 |url=}}</ref> including:
**Finitely presented simple groups.
*Finitely presented [[residually finite]] groups
*One relator groups<ref>{{citation |first1=Roger C. |last1=Lyndon |first2=Paul E |last2=Schupp |title=Combinatorial Group Theory |publisher=Springer |date=2001 |isbn=9783540411581 |pages=1–60 |url=https://books.google.com/books?id=aiPVBygHi_oC&pg=PP1}}</ref> (this is a theorem of Magnus), including:
**Fundamental groups of closed orientable two-dimensional manifolds.
*Combable groups
*Autostackable groups

Examples with unsolvable word problems are also known:
*Given a recursively enumerable set <math>A</math> of positive integers that has insoluble membership problem, <math>\langle a, b, c, d \, | \, a^n b a^n = c^n d c^n : n \in A \rangle</math> is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble{{sfn|Collins|Zieschang|1993|p=149}}
*Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem{{sfn|Collins|Zieschang|1993|loc=Cor. 7.2.6}}
*The number of relators in a finitely presented group with insoluble word problem may be as low as 14 {{sfn|Collins|1969}} or even 12.{{sfn|Borisov|1969}}{{sfn|Collins|1972}}
*An explicit example of a reasonable short presentation with insoluble word problem is given in Collins 1986:{{sfn|Collins|1986}}<ref>We use the corrected version from [http://shell.cas.usf.edu/~eclark/algctlg/groups.html John Pedersen's A Catalogue of Algebraic Systems]</ref>
:<math>\begin{array}{lllll}\langle & a,b,c,d,e,p,q,r,t,k & | & &\\ 
&p^{10}a = ap,  &pacqr = rpcaq,             &ra=ar, &\\
&p^{10}b = bp,  &p^2adq^2r = rp^2daq^2,     &rb=br, &\\
&p^{10}c = cp,  &p^3bcq^3r = rp^3cbq^3,     &rc=cr, &\\
&p^{10}d = dp,  &p^4bdq^4r = rp^4dbq^4,     &rd=dr, &\\
&p^{10}e = ep,  &p^5ceq^5r = rp^5ecaq^5,    &re=er, &\\
&aq^{10} = qa,  &p^6deq^6r = rp^6edbq^6,    &pt=tp, &\\
&bq^{10} = qb,  &p^7cdcq^7r = rp^7cdceq^7,  &qt=tq, &\\
&cq^{10} = qc,  &p^8ca^3q^8r = rp^8a^3q^8,  &&\\
&dq^{10} = qd,  &p^9da^3q^9r = rp^9a^3q^9,  &&\\
&eq^{10} = qe,  &a^{-3}ta^3k = ka^{-3}ta^3  &&\rangle \end{array}</math>