The key idea in geometric group theory is to study infinite groups by endowing them with a
metric and treating them as geometric spaces. This applies to many groups naturally …

We show that for any sequence $ f:{\bf N}\to\{-1,+ 1\} $ taking values in $\{-1,+ 1\} $, the
discrepancy $$\sup_ {n, d\in {\bf N}}\left|\sum_ {j= 1}^ nf (jd)\right| $$ of $ f $ is infinite. This …

In this article we survey recent progress on quasi-isometric rigidity of polycyclic groups.
These results are contributions to Gromov's program for classifying finitely generated groups …

This book provides a general framework for doing geometric group theory for many non-
locally-compact topological transformation groups that arise in mathematical practice …

We give a solution to the inverse problem (given a prescribed function, find a corresponding
group) for large classes of speed, entropy, isoperimetric profile, return probability and L_p …

Coarse differentiation of quasi-isometries II: Rigidity for Sol and lamplighter groups Page 1
Annals of Mathematics 177 (2013), 869-910 http: / / dx.doi.org/10.4007/ annals.2013.177.3.2 …

We build an analogue of the Gromov boundary for any proper geodesic metric space, hence
for any finitely generated group. More precisely, for any proper geodesic metric space $ X …

We study geometric and topological properties of infinite graphs that are quasi-isometric to a
planar graph of bounded degree. We prove that every locally finite quasi-transitive graph …

Let T1,..., Td be homogeneous trees with degrees q1+ 1,..., qd+ 1≥ 3, respectively. For each
tree, let h: Tj→ Z be the Busemann function with respect to a fixed boundary point (end). Its …

In this article, we introduce halo products as a natural generalisation of wreath products.
They also encompass lampshuffler groups $\mathrm {FSym}(H)\rtimes H $ and lampcloner …