Mark Sapir (Vanderbilt and Illinois Math) Flat submaps in CAT(0) $(p,q)$maps and maps with angles Abstract: This is a joint work with A. Olshanskii. Let $p, q$ be positive integers with $1/p+1/q=1/2$. We prove that if a $(p,q)$map $M$ does not contain flat submaps of radius $\ge r$, then its area does not exceed $c(r+1)n$ where $n$ is the perimeter of $M$ and $c$ is an absolute constant. Earlier Ivanov and Schupp proved an exponential bound in terms of $r$. We prove an estimate similar to Ivanov and Schupp for much more general ``maps with angles" which include, for example, van Kampen diagrams over the presentation of the BaumslagSolitar group $BS(1,2)$ and many groups corresponding to $S$machines. We also show that a $(p,q)$ map $M$ tessellating a plane ${\mathbb R}^2$ has path metric quasiisometric to the Euclidean metric on the plane if and only if $M$ has only finitely many nonflat vertices and faces. 
