We will explore the geometry of solvable groups such as lamplighters groups, Baumslag-Solitar groups, Baumslag's metabelian group, and other related examples. We will seek to mix the traditional combinatorial approach with a geometric one. The topic we survey may include: presentations, horocyclic products, lattices, growth, Word and Conjugacy Problems, finiteness properties and Bieri-Neumann-Strebel invariants, quasi-isometric classification and rigidity, and random walks.

General

K. A. Bencsath, M. C. Bonanome, M. H. Dean, M. Zyman, Lectures on Finitely Generated Solvable Groups

C. Drutu's course

J. C. Lennox, D. J. S. Robinson, The Theory of Infinite Soluble Groups

G. Pete, Probability and Geometry on Groups

D. Segal, Polycyclic groups

B.A.F. Wehrfritz, Group and Ring Theoretic Properties of Polycyclic Groups

Nilpotent groups

G. Baumslag, C. F. Miller III, H. Short, Isoperimetric inequalities and the homology of groups (Postscript)

N. Brady, T. R. Riley, H. Short, The Geometry of the Word Problem for Finitely Generated Groups

E. Breuillard, Geometry of locally compact groups of polynomial growth and the shape of large balls

S. M. Gersten, D. F. Holt, T. R. Riley Isoperimetric inequalities for nilpotent groups

P. Pansu's thesis

P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un, Annals of Math., Vol. 129 (1989), p. 1-60.

R. Young's papers

L. van den Dries and A. J. Wilkie, Gromov's Theorem on Groups of Polynomial Growth and Elementary Logic

S. Wenger, Nilpotent groups without exactly polynomial Dehn function

Growth

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc., Vol. 25 (1972), p.603-614.

E. Breuillard, On Uniform Exponential Growth for Solvable Groups

Gromov's polynomial growth theorem

B. Kleiner, A new proof of Gromov's theorem on groups of polynomial growth

T. Tao, A proof of Gromov's Theorem

Milnor and Wolf

Bieri–Neumann–Strebel invariants

R. Strebel, Notes on the Sigma-invariants,

Strebel, Finitely presented soluble groups, in Group theory: Essays for Philip Hall

P. H. Kropholler and J. Mullaney, Cohomological finiteness conditions for a class of metabelian groups

Lamplighter, Baumslag's Group, Diestel–Leader graphs etc.

T.R. Riley and M. Amchislavska, Lamplighter groups and horocyclic products of trees, in preparation

L. Bartholdi, M. Neuhauser, W. Woess, Horocyclic products of trees

G. Baumslag, A finitely presented metabelian group with a free abelian derived group of infinite rank

G. Baumslag, Finitely presented metabelian groups

S. Cleary and T. R. Riley, A finitely presented group with unbounded dead end depth

Y. de Cornulier and R. Tessera. Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups

T.R. Riley and A. Warshall, The unbounded dead end depth property is not a group invariant

W. Woess, Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions

The Tits Alternative

J. Tits, Free subgroups in linear groups, Journal of Algebra 20 (2): 250–270, 1972

The Geometry of Baumslag–Solitar Groups

G. Baumslag, D. Solitar, "Some two generator one-relator non-Hopfian groups" Bull. Amer. Math. Soc. , 689 (1962) pp. 199–201

A. Bendikov, L. Saloff-Coste, M. Salvatori, W. Woess, Brownian motion on treebolic space: escape to infinity

Growth in Baumslag-Solitar groups I: subgroups and rationality, E. Freden, T. Knudson and J. Schoeld

Baumslag–Solitar group, Encyclopedia of Mathematics

B. Farb, L. Mosher, A rigidity theorem for the solvable Baumslag–Solitar groups (With an appendix by Daryl Cooper), Invent. Math. , 131 (1998) pp. 419-451

B. Farb, L. Mosher, Quasi-isometric rigidity for the solvable Baumslag–Solitar groups II, Invent. Math. , 137 (1999) pp. 613-649

A film exploring BS(2,3) by Jürn Laun

K. Whyte, The Geometry of the Higher Baumslag-Solitar Groups

The Geometry of Sol

Notes by Will Dison

Rigidity and q.i. classification

A. Dioubina, Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups

T. Dymarz, Large scale geometry of certain solvable groups

A. Eskin, D. Fisher, K. Whyte, Coarse differentiation of quasi-isometries I & II: Spaces not quasi-isometric to Cayley graphs; Rigidity for Sol and Lamplighter groups

B. Farb and L. Mosher, Problems on the geometry of finitely generated solvable groups

B. Farb and L. Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups, appendix by D. Cooper, Inventiones Math., Vol. 131, No. 2 (1998), p. 419-451.

B. Farb and L. Mosher, On the asymptotic geometry of abelianby-cyclic groups, Acta Math

I. Peng, Coarse differentiation and quasi-isometries of a class of solvable Lie groups I and II

The Geometry of the Word Problem

O. Kharlampovich, A finitely presented solvable group with unsolvable word problem, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 4, 852-873

The Geometry of the Conjugacy Problem

Andrew Sale's papers

M. R. Bridson, T. R. Riley, A. Sale, Survey in preparation

Random walks

Kaimanovich, Poisson boundaries of random walks on discrete solvable groups, Probability measures on groups, X (Oberwolfach, 1990), 205–238, Plenum, New York, 1991

L. Saloff-Coste's papers such as this survey

R. Thompson's papers

W. Woess' papers

Here are some suggestions of presentation topics.

• The Kleiner-Tao-Shalom proof of Gromov's polynomial growth theorem

• The geometry of Baumslag-Solitar groups

• The geometry of Sol

• M. Amchislavska and TRR, The geometry of lamplighter groups

• A. Sale?, The geometry of free solvable groups

• Bieri-Strebel invariants

• Baumslag's embedding theorem for finitely generated metabelian groups

• The geometry of polycyclic groups — I'm not sure what the best way into this topic is. Taking a combinatorial point-of-view, we could look at presentations of polycyclic groups and collection strategies — e.g. Chapter 8 of The Handbook of Computational Group Theory by Holt, Eick and O'Brien. Taking a geometric point-of-view, there's this talk of Eskin.

• The Tits Alternative

• A survey of rigidity in solvable groups

• J. Cuno + ??, Random walks, Poisson boundaries, rate of escape on solvable groups

• Properly Discontinuous Groups of Affine Transformations; Auslander's conjecture

• Amin Saied, The complete course notes

• Amin Saied, Finitely presented metabalian groups

• Teddy Einstein, Sigma Invariants