Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is as such unique. Our approach provides a different proof for the analog situation in ${\mathbb H^2}$ where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

In this work, we investigate how to decompose a pair $(A,B)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\langle A,B\rangle$.

Let $\Gamma \subset \SO(3,1)$ be a lattice. The well known \emph{bending deformations}, introduced by \linebreak Thurston and Apanasov, can be used to construct non-trivial curves of representations of $\Gamma$ into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in $\H^1(\Gamma, \mink{4})$. Our main result generalizes this construction of cohomology to the context of ``branched'' totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in $S^3$ which is not infinitesimally rigid in $\SO(4,1)$. The first order deformations of this link complement are supported on a piecewise totally geodesic $2$-complex.

We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$.

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)$-space $\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of $\mbox{H}^{n+1}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\PSL_2 (\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.

Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme.

We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Ga$-automorphic Laplacian below a given bound when $\Ga$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Ga$ in the family is assumed to contain a principal congruence subgroup whose index in $\Ga$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Ga$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice~$\Ga$.