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The trivial group has exactly one element, its identity. Every group homomorphism to it is trivial, and a group is trivial precisely when all elements in a generating set equal the identity.

The trivial representation sends every group element to the identity linear map.

A group generated by two involutions is dihedral in the extended convention allowing $C_2$ and the infinite dihedral group. If $a\ne b$, put $x=ab$, $y=b$; then $yx{y}^{-1}=x^{-1}$. Conversely, from the inversion relation, $y$ and $xy$ square to the identity and generate. If $xy=e$, the group is $C_2$ and the same nonidentity element is used twice. Exact order two, rather than merely a square equal to the identity, requires this exceptional case. This convention extends the usual finite dihedral group definition.

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 $f$.

The complex upper half-plane is $\mathbb{H}=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$.

A group is virtually solvable, or virtually soluble, when it contains a solvable group as a finite-index subgroup. This is preserved under subgroups, quotients and group extensions. The extension proof can use a finite-index characteristic soluble subgroup of the kernel, followed by centralizing the resulting finite normal kernel. Finite generation is not required for these closure properties.

The word length $|g{|}_S$ is the least number of letters from a symmetric generating set of a group $S$ whose product is $g$. The empty word has length zero. It obeys $|gh{|}_S\le |g{|}_S+|h{|}_S$, $|g^{-1}{|}_S=|g{|}_S$ and $d_S(g,h)=|g^{-1}h{|}_S$. The generating set may be infinite; finiteness is needed only for additional conclusions such as finite metric balls.

The word metric associated with a generating set $S$ is the path metric on ${\mathrm{Cay}}_S(G)$, equivalently $d_S(g,h)=|g^{-1}h{|}_S$.