** "Geometry and Analysis on Groups" Research Seminar **

**Time:** 2016.06.08, 13:00–15:00

**Location:** 2.Stock Besprechungsraum, Oskar-Morgenstern-Platz 1

**Title:** "Classifying spaces for families of
subgroups for systolic groups."

**Speaker:** Damian Osajda
(Uniwersytet Wrocławski)

**Abstract:**
This is joint work with Tomasz Prytula (Kopenhagen). For a given group \(G\)
a family of its subgroups is a collection of subgroups which is closed under taking subgroups
and conjugations by elements of \(G\). Examples are: the family consisting of only
the trivial group, the family of finite subgroups, the family of virtually cyclic subgroups.
A classifying space for a given family is a \(G\)-\(CW\)–complex with stabilizers in the family and such
that the fixed point set of any subgroup in the family is contractible. Our main result is a
construction of low-dimensional classifying spaces for the family of virtually cyclic subgroups
for groups acting properly on systolic complexes. Such spaces are of interest in view of their
appearance on the left-hand side of the assembly map in the formulation of the Farrell-Jones
conjecture. The main technical result is the determination of the large-scale structure
of the minimal displacement set of a hyperbolic isometry of a systolic complex. As an application
one gets a construction of corresponding low-dimensional classifying spaces for groups acting
properly on graphical small cancellation complexes. I will mention other applications.