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Maximizing the Gibbs entropy under normalization alone gives a uniform distribution. Adding a fixed mean-energy constraint gives the Boltzmann distribution $p_n=Z^{-1}{e}^{-\beta E_n}$.

An equatorial wave is trapped near the equator by the sign change of the Coriolis parameter. Its meridional structures are Gaussian-weighted Hermite polynomials.

A Rossby wave is a low-frequency wave restored by the spatial variation of planetary or background potential vorticity. On a beta plane its phase commonly propagates westward.

Ocean circulation comprises wind-driven, buoyancy-driven, and tidal motions of the ocean over a wide range of spatial and temporal scales.

Geostrophic adjustment is the radiation of inertia-gravity waves from an unbalanced disturbance, leaving a slower flow in geostrophic balance while conserving linearized potential vorticity.

The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of potential vorticity.

A rotating fluid is described in a rotating reference frame, where the Coriolis acceleration and centrifugal acceleration supplement the physical forces.

The Boussinesq approximation treats density as constant except where a small density variation multiplies gravity to produce buoyancy. The velocity remains incompressible.

An action is properly discontinuous when every compact set meets only finitely many of its translates. For an isometric action on a proper metric space, this gives the finiteness needed to extract a constant group element from a sequence with bounded source and target.

For a discrete group action on a locally compact space, a standard proper-discontinuity condition is that only finitely many group elements translate a given compact set to meet itself.

A group action on a space is cocompact when its quotient is compact, equivalently when some compact set has translates covering the space.

A metric space is proper when every closed bounded subset is compact.

For an isometry $g$ of a metric space, its displacement function is $d_g(x)=d(x,gx)$. It is $2$-Lipschitz.

A subset $Y$ of a geodesic metric space is $K$-quasiconvex when every metric geodesic between points of $Y$ lies in the closed $K$-neighborhood of $Y$.

A map $f:X\to Y$ is a $(\lambda ,\epsilon )$-quasi-isometric embedding whenIt is a quasi-isometry when every point of $Y$ lies a uniformly bounded distance from its image.

A Fuchsian group is a discrete subgroup of ${\mathrm{PSL}}_2(\mathbb{R})$, acting by orientation-preserving isometries of the hyperbolic plane. It is non-elementary when it has no finite orbit in the hyperbolic plane or its ideal boundary.

A group action on a tree is a homomorphism to the tree's combinatorial isometry group. Barycentric subdivision removes edge inversions without changing the essential action.

Bass-Serre theory describes group splittings as actions on trees and reconstructs a graph of groups from the stabilizers of such an action.

A covering graph is a graph map that induces a bijection on the incident oriented edges at every vertex. Connected based covers of a graph correspond to subgroups of its fundamental group.

If a group has a properly discontinuous group action by isometries that is also a cocompact group action on a nonempty proper metric space that is a geodesic metric space, then the group is finitely generated and every orbit map from the word metric for any finite generating set of a group is a quasi-isometry. Finite stabilizers are permitted.

A Cayley graph turns multiplication in a group into travel: the chosen generators of a group are the allowed one-step moves, and a word over an alphabet becomes a route. This lets us study a finitely generated group using graph distances, closed balls, and geodesics, rather than only equations between products.

Let $G$ be a group with a finite generating set of a group $S$ satisfying $S=S^{-1}$ and $1\notin S$. Such a symmetric subset of a group includes the inverse element of every allowed move. The Cayley graph ${\mathrm{Cay}}_S(G)$ has vertex set $G$ and joins $g$ to $gs$ for each $s\in S$. We choose right multiplication for these moves. Reversing the move uses $s^{-1}$, so for graph distance we regard $g\to gs$ and $gs\to g$ as the two orientations of one undirected edge. An involution therefore gives one edge, not two parallel edges. Each edge has length one, regardless of its length in a drawing.^[ref]

A sequence $s_1,\dots ,s_n$ labels the walkIt ends at $h$ exactly when $g^{-1}h=s_1\cdots s_n$. Thus the Cayley graph is a connected graph precisely because $S$ generates $G$, and its graph distance on vertices is the word metric. Left multiplication by $a$ sends the edge from $g$ to $gs$ to the edge from $ag$ to $(ag)s$, preserving its generator label: this is a group action by graph automorphisms and isometries. Viewing groups as symmetries of geometric objects is the motivation of Clay and Margalit's introduction.^[ref]

The useful geometric object includes its allowed moves, not just its drawing. Cayley graphs of the integers show how changing the moves adds shortcuts; word metric balls in the integer lattice show how the same group can acquire visibly different closed balls. Meier's book develops this passage from group operations to graphs as its central motivation.^[ref]