Universitat Wien


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  for Research - IDEAS
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  European Research Council
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limits, curvature, and randomness

The ERC grant of Goulnara ARZHANTSEVA: ANALYTIC no. 259527
Start date: 2011-04-01
End date: 2016-03-31


The overall goal of this project is to develop new concepts and techniques in geometric and asymptotic group theory for a systematic study of the analytic properties of discrete groups. These are properties depending on the unitary representation theory of the group. The fundamental examples are amenability, discovered by von Neumann in 1929, and property (T), introduced by Kazhdan in 1967. My main objective is to establish the precise relations between groups recently appeared in K-theory and topology such as C*-exact groups and groups coarsely embeddable into a Hilbert space, versus those discovered in ergodic theory and operator algebra, for example, sofic and hyperlinear groups. This is a first ever attempt to confront the analytic behaviour of so different nature. I plan to work on crucial open questions: Is every coarsely embeddable group C*-exact? Is every group sofic? Is every hyperlinear group sofic? My motivation is two-fold: - Many outstanding conjectures were recently solved for these groups, e.g. the Novikov conjecture (1965) for coarsely embeddable groups by Yu in 2000 and the Gottschalk surjunctivity conjecture (1973) for sofic groups by Gromov in 1999. However, their group-theoretical structure remains mysterious. - In recent years, geometric group theory has undergone significant changes, mainly due to the growing impact of this theory on other branches of mathematics. However, the interplay between geometric, asymptotic, and analytic group properties has not yet been fully understood. The main innovative contribution of this proposal lies in the interaction between 3 axes: (i) limits of groups, in the space of marked groups or metric ultralimits; (ii) analytic properties of groups with curvature, of lacunary or relatively hyperbolic groups; (iii) random groups, in a topological or statistical meaning. As a result, I will describe the above apparently unrelated classes of groups in a unified way and will detail their algebraic behaviour.


Infinite monster groups

Erwin Schrödinger International Institute, Vienna
Program: December 12 - 21, 2011 / Workshop: December 14 - 18, 2011

Word maps and stability of representations

Erwin Schrödinger International Institute, Vienna
April 11-12-13, 2013

Geometry of computation in groups

Erwin Schrödinger International Institute, Vienna
Research activities: March 31 - April 13, 2014 Workshop: March 31 - April 4, 2014


Geometry and Analysis on Groups

meets weekly


Geometric and Asymptotic Group Theory


Team Members: Other members (financed by other sources but contributing to the research): Open Positions


Publications and preprints supported by the grant.
For additional works in progress during this period, see the web pages of the individual team members.

Vadim Alekseev and Martin Finn-Sell, Sofic boundaries of groups and coarse geometry of sofic approximations, preprint, arXiv:1608.02242.

Vadim Alekseev and Martin Finn-Sell, Non-amenable principal groupoids with weak containment, International Mathematical Research Notices (in press).

Goulnara Arzhantseva, Asymptotic approximations of finitely generated groups, Extended abstracts Fall 2012---automorphisms of free groups, Trends Math. Res. Perspect. CRM Barc., vol. 1, Springer, Cham, 2014, pp. 7--15.

Goulnara Arzhantseva and Arindam Biswas, Large girth graphs with bounded diameter-by-girth ratio, preprint, arXiv:1803.09229.

Goulnara Arzhantseva and Cornelia Drutu, Geometry of infinitely presented small cancellation groups, rapid decay and quasi-homomorphisms, preprint, arXiv:1212.5280.

Goulnara Arzhantseva and Światoslaw R. Gal, On approximation properties of semidirect products of groups, preprint, arXiv:1312.7682.

Goulnara Arzhantseva and Erik Guentner, Coarse non-amenability and covers with small eigenvalues, Mathematische Annalen 354 (2012), no. 3, 863--870.

Goulnara Arzhantseva, Erik Guentner, and Ján Špakula, Coarse non-amenability and coarse embeddings, Geometric and Functional Analysis [GAFA] 22 (2012), no. 1, 22--36.

Goulnara Arzhantseva, Jean-François Lafont, and Ashot Minasyan, Isomorphism versus commensurability for a class of finitely presented groups, J. Group Theory 17 (2014), no. 2, 361--378.

Goulnara Arzhantseva, Graham A. Niblo, Nick Wright, and Jiawen Zhang, A characterization for asymptotic dimension growth, Algebr. Geom. Topol. 18 (2018), no. 1, 493--524.

Goulnara Arzhantseva and Damian Osajda, Graphical small cancellation groups with the Haagerup property, preprint, arXiv:1404.6807.

Goulnara Arzhantseva and Damian Osajda, Infinitely presented small cancellation groups have the Haagerup property, Journal of Topology and Analysis 7 (2015), no. 3, 389--406.

Goulnara Arzhantseva and Liviu Păunescu, Almost commuting permutations are near commuting permutations, Journal of Functional Analysis 269 (2015), no. 3, 745--757.

Goulnara Arzhantseva and Liviu P~aunescu, Linear sofic groups and algebras, Trans. Amer. Math. Soc. 369 (2017), no. 4, 2285--2310.

Goulnara Arzhantseva and Markus Steenbock, Rips construction without unique product, preprint, arXiv:1407.2441.

Goulnara Arzhantseva and Romain Tessera, Relative expanders, Geometric and Functional Analysis [GAFA] 25 (2015), no. 2, 317--341.

Goulnara N. Arzhantseva, Christopher H. Cashen, Dominik Gruber, and David Hume, Negative curvature in graphical small cancellation groups, preprint, arXiv:1602.03767.

Goulnara N. Arzhantseva, Christopher H. Cashen, Dominik Gruber, and David Hume, Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction, Doc. Math. 22 (2017), 1193--1224.

Goulnara N. Arzhantseva, Christopher H. Cashen, and Jing Tao, Growth tight actions, Pacific Journal of Mathematics 278 (2015), no. 1, 1--49.

Goulnara N. Arzhantseva, Andreas Thom, and Alain Valette, Finite-dimensional approximations of discrete groups, Oberwolfach Reports 8 (2011), no. 2, 1429--1467.

Federico Berlai, Groups satisfying Kaplansky's stable finiteness conjecture, preprint, arXiv:1501.02893.

Federico Berlai, Residual properties of free products, Comm. Algebra 44 (2016), no. 7, 2959--2980.

Federico Berlai, Dikran Dikranjan, and Anna Giordano Bruno, Scale function vs topological entropy, Topology and its Applications 160 (2013), no. 18, 2314--2334.

Federico Berlai and Michal Ferov, Residual properties of graph products of groups, J. Group Theory 19 (2016), no. 2, 217--231.

Christopher H. Cashen, Splitting line patterns in free groups, Algebr. Geom. Topol. 16 (2016), no. 2, 621--673.

Christopher H. Cashen, A geometric proof of the structure theorem for cyclic splittings of free groups, Topology Proc. 50 (2017), 335--349.

Christopher H. Cashen and Gilbert Levitt, Mapping tori of free group automorphisms, and the Bieri-Neumann-Strebel invariant of graphs of groups, J. Group Theory 19 (2016), no. 2, 191--216.

Christopher H. Cashen and Jason F. Manning, Virtual geometricity is rare, LMS J. Comput. Math. 18 (2015), no. 1, 444--455.

Christopher H. Cashen and Alexandre Martin, Quasi-isometries between groups with two-ended splittings, Math. Proc. Cambridge Philos. Soc. 162 (2017), no. 2, 249--291.

Christopher H. Cashen and Jing Tao, Growth tight actions of product groups, Groups Geom. Dyn. 10 (2016), no. 2, 753--770.

Jérémie Chalopin, Victor Chepoi, and Damian Osajda, On two conjectures of Maurer concerning basis graphs of matroids, Journal of Combinatorial Theory, Series B 114 (2015), 1--32.

Mauro Di Nasso, Isaac Goldbring, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg, High density piecewise syndeticity of product sets in amenable groups, J. Symb. Log. 81 (2016), no. 4, 1555--1562.

Martin Finn-Sell, Almost quasi-isometries and more non-C*-exact groups, Math. Proc. Cambridge Philos. Soc. 162 (2017), no. 3, 393--403.

Martin Finn-Sell and Jianchao Wu, The asymptotic dimension of box spaces for elementary amenable groups, preprint, arXiv:1508.05018.

Światoslaw R. Gal and Jarek Kedra, On bi-invariant word metrics, Journal of Topology and Analysis 3 (2011), no. 2, 161--175.

Isaac Goldbring and Martino Lupini, Model-theoretic aspects of the gurarij operator system, preprint, arXiv:1501.04332.

Roger Gómez-Ortells, Compactly supported cohomology of systolic 3-pseudomanifolds, Colloquium Mathematicum 135 (2014), no. 1, 103--112.

Dominik Gruber, Groups with graphical C(6) and C(7) small cancellation presentations, Transactions of the American Mathematical Society 367 (2015), no. 3, 2051--2078.

Dominik Gruber, Infinitely presented C(6)-groups are SQ-universal, Journal of the London Mathematical Society (2) 92 (2015), no. 1, 178--201.

Dominik Gruber, Alexandre Martin, and Markus Steenbock, Finite index subgroups without unique product in graphical small cancellation groups, Bulletin of the London Mathematical Society 47 (2015), no. 4, 631--638.

Dominik Gruber and Alessandro Sisto, Infinitely presented graphical small cancellation groups are acylindrically hyperbolic, Ann. Inst. Fourier (in press), preprint.

Michael Hartz and Martino Lupini, The classification problem for operator algebraic varieties and their multiplier algebras, Trans. Amer. Math. Soc. 370 (2018), no. 3, 2161--2180.

Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, The Australasian Journal of Combinatorics 52 (2012), 41--54.

Jesús Leaños, C. Ndatchi-Mbe-Koua, and Luis Manuel Rivera-Martínez, Euclidean arrangements of n pseudolines with one n-gon are stretchable, Proceedings of the XIV Spanish Meeting on Computational Geometry, vol. 8, 2011, pp. 129--132.

Martino Lupini, Fraïssé limits in functional analysis, preprint, arXiv:1510.05188.

Alexandre Martin, Acylindrical actions on CAT(0) square complexes, preprint, arXiv:1509.03131.

Alexandre Martin, Combination of universal spaces for proper actions, J. Homotopy Relat. Struct. 10 (2015), no. 4, 803--820.

Alexandre Martin, Complexes of groups and geometric small cancelation over graphs of groups, Bull. Soc. Math. France 145 (2017), no. 2, 193--223.

Alexandre Martin, On the cubical geometry of Higman's group, Duke Math. J. 166 (2017), no. 4, 707--738.

Alexandre Martin, Combination of classifying spaces for proper actions, Journal of Homotopy Theory and Related Structures (in press).

Alexandre Martin and Markus Steenbock, Cubulation of small cancellation groups over free products, preprint, arXiv:1409.3678.

Alexandre Martin and Markus Steenbock, A combination theorem for cubulation in small cancellation theory over free products, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 4, 1613--1670.

Alexandre Martin and Jacek Światkowski, Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries, J. Group Theory 18 (2015), no. 2, 273--289.

Rutilo Moreno and Luis Manuel Rivera, Blocks in cycles and k-commuting permutations, SpringerPlus 5 (2016), no. 1, 1949.

Damian Osajda, Small cancellation labellings of some infinite graphs and applications, preprint (2014), arXiv:1406.5015.

Damian Osajda, Combinatorial negative curvature and triangulations of three-manifolds, Indiana University Mathematics Journal 64 (2015), no. 3, 943--956.

Markus Steenbock, Rips-Segev torsion-free groups without the unique product property, Journal of Algebra 438 (2015), 337--378.

Rudolf Zeidler, Coarse median structures and homomorphisms from Kazhdan groups, Geometriae Dedicata (2015), 1--20.

Jiawen Zhang, Equi-variant and stable finite decomposition complexity, preprint, arXiv:1509.08883.


Fakultät für Mathematik
Universität Wien
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria