Combinatorial group theory

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, combinatorial group theory is the theory of <a href="/facts/Free_group/zPo2NIGb">free groups</a>, and the concept of a <a href="/facts/Presentation_of_a_group/f9emPUoI">presentation of a group</a> by <a href="/facts/Generator_(mathematics)/Jo1eBRbD">generators</a> and <a href="/facts/Relation_(mathematics)/HjblDaMy">relations</a>. It is much used in <a href="/facts/Geometric_topology/NpsqSK2N">geometric topology</a>, the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of a <a href="/facts/Simplicial_complex/fF2282CF">simplicial complex</a> having in a natural and geometric way such a presentation.
A very closely related topic is <a href="/facts/Geometric_group_theory/AJhxpFUe">geometric group theory</a>, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.
</p><p>It also comprises a number of <a href="/facts/Algorithmically_insoluble/QBxmXBbY">algorithmically insoluble</a> problems, most notably the <a href="/facts/Word_problem_for_groups/eKBrkBPv">word problem for groups</a>; and the classical <a href="/facts/Burnside_problem/RsWB8d31">Burnside problem</a>.
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Combinatorial group theory open-in-new

Combinatorial group theory