The large-scale geometry of locally compact solvable groups originated from the works of M. Gromov in the 80’s and early 90’s. The main attention is focused on invariants for finitely generated groups. The paper under review is devoted to a survey of three types of geometric invariants: the probability of return of symmetric random walks, the cohomological properties as geometric invariants for amenable groups and Dehn functions of solvable groups.
Let \(G\) be a locally compact, compactly generated group, \(\mu\) be a Borel probability measure on \(G\), which can be interpreted as the probability transition of random walk on \(G\) by a process \((X_n)_{n\geq 0}\) starting at the neutral element \(X_0=1_G\), \(X_{n+1} = X_n\xi_n\), where \((\xi_n)_{n\geq 0}\) is an i.i.d. sequence of \(G\)-valued random variables of distribution \(\mu\). The distribution of \(X_n\) is the \(n\)-fold convolution product \(\mu^{(n)} = \mu * \dots * \mu\). Assume that \(\mu\) is symmetric (i.e., \(\mu(U) = \mu(U^{-1})\) for every Borel set \(U\)) and is absolutely continuous with respect to a fixed Haar measure \(\lambda\) on \(G\), that its support is compact and generates \(G\) and that the density of \(\mu\) is continuous. Define the invariant \(\phi(n) = \frac{d\mu^{(2n)}}{d\lambda}(1)\). Following Kesten \(G\) is nonamenable if and only if \(\phi(n)\preceq e^{-n}\). Some interesting results of this invariant \(\phi(n)\) are: the upper bound \(\phi(n) \preceq n^{-D/2}\) is equivalent to the Sobolev inequality \(||f||_{\frac{2D}{D-2}} \leq C ||\nabla f||_2\) (Theorem 3.1). Let \(G\) be a nilpotent connected Lie group of volume growth exponent \(D= \sum_{i\geq 1} i\dim(C^i(G)/C^{i+1}(G))\) for the lower central series \(C^i(G)\), then \(\phi(n) \approx n^{-\frac{D}{2}}\) (Theorem 3.2).
Let \(G\) be a compact second countable group, and \(\pi\) a continuous unitary representation on a Hilbert space \(\mathcal H\). For any \(n\geq 0\) one defines \(n\)-cohomology \(H^n(G,\pi) = Z(G,\pi)/B(G,\pi)\) and reduced \(n\)-cohomology \(\bar{H}^n(G,\pi) = Z(G,\pi)/\overline{B(G,\pi)}\), and says that it has property \(H_{FD}\) (resp. \(H_T\)) if for every unitary representation \(\pi\) with \(\bar{H}^1(G,\pi) \neq 0\), there exists a nonzero subrepresentation of finite dimension (resp. nonzero trivial subrepresentation). Some interesting results related with this cohomological invariant are surveyed: the cohomological dimension over \(\mathbb Q\) is stable under quasi-isometry among the class of all amenable groups (Theorem 4.8) or the Betti numbers are invariant under quasi-isometry among nilpotent groups (Theorem 4.9). If \(\Lambda\) and \(\Gamma\) are quasi-isometric nilpotent groups, then the real cohomology ring \(H^*(\Lambda, \mathbb R)\) and \(H^*(\Gamma,\mathbb R)\) are isomorphic as graded rings (Theorem 4.10).
Finally the Dehn function of solvable groups is introduced as \[ \delta(n) = \sup\{area(w): w \text{ relates of length } \leq n\}, \] where the area of a word \(w\) in the letters of a compact generating subset of a locally compact compactly generated group \(G\) is a product of \(\leq n\) conjugates of relations of lenght \(\leq k \in \mathbb N\). This invariant plays a crucial role in the theory of solvable groups (Theorems 5.5, 5.6, 5.7,5.9, 5.10).
The survey is understandable and clearly presented.