\documentclass[8pt]{beamer}
\usepackage[ansinew]{inputenc}
\usepackage{amssymb, color}
\usepackage{graphicx}
\usepackage{pgf,tikz}
\usepackage{mathrsfs}
\usetikzlibrary{arrows}
\usepackage{beamerthemesplit}
%\usepackage[french]{babel}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{prop}[lem]{Proposition}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{defi}[lem]{Definition}
\newtheorem{rem}[lem]{Remark}
\newtheorem{nota}[lem]{Notation}
\newtheorem{ques}[lem]{Question}
\newtheorem{prob}[lem]{Problem}
\newtheorem{sol}[lem]{Solution}
\newtheorem{conc}[lem]{Conclusion}
\newtheorem{conj}[lem]{Conjecture}
\newtheorem{ex}[lem]{Example}
\newtheorem{exs}[lem]{Examples}
\newtheorem{res}[lem]{Results}
\newtheorem{fac}[lem]{Fact}
\newtheorem{plan}[lem]{Plan of lecture}
\newtheorem{scheme}[lem]{Scheme of proof}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\n#1{{|\hskip-1pt | #1 |\hskip-1pt |}}



\title{Persistence in geometric group theory}
\author{Pierre Pansu, Universit\'e Paris-Sud/Paris-Saclay}
\date{September 5th, 2019}


\mode<presentation>{\usetheme{Warsaw}}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}
\begin{center}
\definecolor{uuuuuu}{rgb}{0.26666666666666666,0.26666666666666666,0.26666666666666666}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.2cm,y=0.2cm]
%\draw [color=cqcqcq,, xstep=0.4cm,ystep=0.4cm] (-21.321804244784126,-15.1389536090264) grid (19.45693282169477,3.1646548863659314);
\clip(-21.321804244784126,-15.1389536090264) rectangle (19.45693282169477,3.1646548863659314);
\draw (0.,-6.)-- (16.,-6.);
\draw (8.,2.)-- (8.,-14.);
\draw (5.139926682871062,-1.9625794616877066)-- (10.964530097651974,-1.9625794616877066);
\draw (5.002337625828993,-9.942744770127693)-- (11.193845192722089,-10.034470808155739);
\draw (3.9933512075204884,-3.0632919180242566)-- (3.9933512075204884,-9.025484389847236);
\draw (11.973516515960478,-3.200880975066325)-- (12.019379534974501,-9.025484389847236);
\draw (3.030227808226007,-4.026415317318738)-- (5.002337625828993,-4.026415317318738);
\draw (5.965461025123474,-0.8160039863371338)-- (6.011324044137497,-2.879839841968165);
\draw (10.001406698357492,-0.9077300243651796)-- (10.001406698357492,-2.788113803940119);
\draw (3.121953846254053,-8.016497971538731)-- (5.139926682871062,-8.016497971538731);
\draw (11.147982173708066,-8.016497971538731)-- (13.028365953283005,-7.970634952524708);
\draw (7.157899519488071,0.835064698167691)-- (9.130009337091057,0.835064698167691);
\draw (7.112036500474048,-13.015567044067229)-- (9.130009337091057,-13.015567044067229);
\draw (5.919598006109451,-8.93375835181919)-- (5.965461025123474,-11.135183264492289);
\draw (9.955543679343469,-9.025484389847236)-- (10.001406698357492,-11.043457226464243);
\draw (1.1039810096370442,-4.989538716613219)-- (1.0581179906230214,-7.007511553230227);
\draw (1.0581179906230214,-7.007511553230227)-- (1.1039810096370442,-4.989538716613219);
\draw (11.102119154694043,-3.934689279290692)-- (13.074228972297028,-3.934689279290692);
\draw (15.046338789900014,-5.035401735627242)-- (15.092201808914037,-6.961648534216204);
\draw (-18.895511955860016,-5.987149361330235)-- (-8.89237708047114,-5.987149361330235);
\draw (-18.93807848724465,-3.986522386252468)-- (-8.934943611855775,-3.986522386252468);
\draw (-18.895511955860016,-1.9858954111747016)-- (-8.89237708047114,-2.0284619425593347);
\draw (-18.895511955860016,-7.987776336408001)-- (-8.89237708047114,-7.987776336408001);
\draw (-18.93807848724465,-9.988403311485767)-- (-8.89237708047114,-9.988403311485767);
\draw (-18.00161479678271,-0.9217321265588683)-- (-18.00161479678271,-11.0525665961016);
\draw (-15.958421290320302,-0.9642986579435016)-- (-16.000987821704935,-11.0525665961016);
\draw (-13.957794315242529,-0.9642986579435016)-- (-13.957794315242529,-10.967433533332335);
\draw (-12.042300402934021,-1.0494317207127684)-- (-11.999733871549386,-10.967433533332335);
\draw (-9.999106896471613,-1.134564783482035)-- (-10.041673427856246,-11.010000064716968);
\begin{scriptsize}
\draw [fill=uuuuuu] (1.081015980469163,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (3.993351207520489,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (3.993351207520489,-4.0264153173187385) circle (1.5pt);
\draw [fill=uuuuuu] (3.993351207520489,-8.01649797153873) circle (1.5pt);
\draw [fill=uuuuuu] (5.990940480131265,-1.9625794616877068) circle (1.5pt);
\draw [fill=uuuuuu] (8.,0.835064698167691) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-1.9625794616877068) circle (1.5pt);
\draw [fill=uuuuuu] (10.00140669835749,-1.9625794616877068) circle (1.5pt);
\draw [fill=uuuuuu] (11.979294534103975,-3.934689279290692) circle (1.5pt);
\draw [fill=uuuuuu] (11.995556823243419,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (15.069305415242221,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (12.01126896651983,-7.9954421961043) circle (1.5pt);
\draw [fill=uuuuuu] (5.940908238778829,-9.956649519949172) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-9.987154583078373) circle (1.5pt);
\draw [fill=uuuuuu] (9.978065835531563,-10.016459262123286) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-13.015567044067225) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-1.9896992288729032) circle (1.5pt);
\draw [fill=uuuuuu] (-18.00161479678271,-3.986522386252468) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-5.9871493613302365) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-7.987776336408002) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-15.99649768126352,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-15.988056217233657,-7.987776336408002) circle (1.5pt);
\draw [fill=uuuuuu] (-15.979614753203792,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-15.971173289173926,-3.9865223862524677) circle (1.5pt);
\draw [fill=uuuuuu] (-15.96278448206971,-1.9983751025525325) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-2.006906975602861) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-7.987776336408002) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-12.00393571799508,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-12.012522099862366,-7.987776336408001) circle (1.5pt);
\draw [fill=uuuuuu] (-12.021108481729653,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-12.02969486359694,-3.986522386252468) circle (1.5pt);
\draw [fill=uuuuuu] (-12.038156008390947,-2.015075649249038) circle (1.5pt);
\draw [fill=uuuuuu] (-10.002939531650116,-2.0237361448947433) circle (1.5pt);
\draw [fill=uuuuuu] (-10.01139981717321,-3.9865223862524677) circle (1.5pt);
\draw [fill=uuuuuu] (-10.020023209307166,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-10.028646601441123,-7.987776336408001) circle (1.5pt);
\draw [fill=uuuuuu] (-10.037269993575077,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-3.986522386252468) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}

%\noindent \hskip-1cm Free abelian group \hskip2.5cm Free group \hskip.5cm{}
\begin{minipage}{5.5cm}
\begin{center}
Free abelian group

$\langle a,b|aba^{-1}b^{-1}\rangle$ 

Fundamental group of 2-torus

\includegraphics[width=1in]{torus.png}
\end{center}
\end{minipage}
\hskip-.91cm
\begin{minipage}{5cm}
\begin{center}
Free group

$\langle a,b|\rangle$ 

Fund. group of punctured 2-torus

\includegraphics[width=1in]{wedge.png}
\end{center}
\end{minipage}

\end{frame}


\begin{frame}

\begin{center}
\definecolor{uuuuuu}{rgb}{0.26666666666666666,0.26666666666666666,0.26666666666666666}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.2cm,y=0.2cm]
%\draw [color=cqcqcq,, xstep=0.4cm,ystep=0.4cm] (-21.321804244784126,-15.1389536090264) grid (19.45693282169477,3.1646548863659314);
\clip(-21.321804244784126,-15.1389536090264) rectangle (19.45693282169477,3.1646548863659314);
\draw (0.,-6.)-- (16.,-6.);
\draw (8.,2.)-- (8.,-14.);
\draw (5.139926682871062,-1.9625794616877066)-- (10.964530097651974,-1.9625794616877066);
\draw (5.002337625828993,-9.942744770127693)-- (11.193845192722089,-10.034470808155739);
\draw (3.9933512075204884,-3.0632919180242566)-- (3.9933512075204884,-9.025484389847236);
\draw (11.973516515960478,-3.200880975066325)-- (12.019379534974501,-9.025484389847236);
\draw (3.030227808226007,-4.026415317318738)-- (5.002337625828993,-4.026415317318738);
\draw (5.965461025123474,-0.8160039863371338)-- (6.011324044137497,-2.879839841968165);
\draw (10.001406698357492,-0.9077300243651796)-- (10.001406698357492,-2.788113803940119);
\draw (3.121953846254053,-8.016497971538731)-- (5.139926682871062,-8.016497971538731);
\draw (11.147982173708066,-8.016497971538731)-- (13.028365953283005,-7.970634952524708);
\draw (7.157899519488071,0.835064698167691)-- (9.130009337091057,0.835064698167691);
\draw (7.112036500474048,-13.015567044067229)-- (9.130009337091057,-13.015567044067229);
\draw (5.919598006109451,-8.93375835181919)-- (5.965461025123474,-11.135183264492289);
\draw (9.955543679343469,-9.025484389847236)-- (10.001406698357492,-11.043457226464243);
\draw (1.1039810096370442,-4.989538716613219)-- (1.0581179906230214,-7.007511553230227);
\draw (1.0581179906230214,-7.007511553230227)-- (1.1039810096370442,-4.989538716613219);
\draw (11.102119154694043,-3.934689279290692)-- (13.074228972297028,-3.934689279290692);
\draw (15.046338789900014,-5.035401735627242)-- (15.092201808914037,-6.961648534216204);
\draw (-18.895511955860016,-5.987149361330235)-- (-8.89237708047114,-5.987149361330235);
\draw (-18.93807848724465,-3.986522386252468)-- (-8.934943611855775,-3.986522386252468);
\draw (-18.895511955860016,-1.9858954111747016)-- (-8.89237708047114,-2.0284619425593347);
\draw (-18.895511955860016,-7.987776336408001)-- (-8.89237708047114,-7.987776336408001);
\draw (-18.93807848724465,-9.988403311485767)-- (-8.89237708047114,-9.988403311485767);
\draw (-18.00161479678271,-0.9217321265588683)-- (-18.00161479678271,-11.0525665961016);
\draw (-15.958421290320302,-0.9642986579435016)-- (-16.000987821704935,-11.0525665961016);
\draw (-13.957794315242529,-0.9642986579435016)-- (-13.957794315242529,-10.967433533332335);
\draw (-12.042300402934021,-1.0494317207127684)-- (-11.999733871549386,-10.967433533332335);
\draw (-9.999106896471613,-1.134564783482035)-- (-10.041673427856246,-11.010000064716968);
\begin{scriptsize}
\draw [fill=uuuuuu] (1.081015980469163,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (3.993351207520489,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (3.993351207520489,-4.0264153173187385) circle (1.5pt);
\draw [fill=uuuuuu] (3.993351207520489,-8.01649797153873) circle (1.5pt);
\draw [fill=uuuuuu] (5.990940480131265,-1.9625794616877068) circle (1.5pt);
\draw [fill=uuuuuu] (8.,0.835064698167691) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-1.9625794616877068) circle (1.5pt);
\draw [fill=uuuuuu] (10.00140669835749,-1.9625794616877068) circle (1.5pt);
\draw [fill=uuuuuu] (11.979294534103975,-3.934689279290692) circle (1.5pt);
\draw [fill=uuuuuu] (11.995556823243419,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (15.069305415242221,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (12.01126896651983,-7.9954421961043) circle (1.5pt);
\draw [fill=uuuuuu] (5.940908238778829,-9.956649519949172) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-9.987154583078373) circle (1.5pt);
\draw [fill=uuuuuu] (9.978065835531563,-10.016459262123286) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-13.015567044067225) circle (1.5pt);
\draw [fill=uuuuuu] (8.,-6.) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-1.9896992288729032) circle (1.5pt);
\draw [fill=uuuuuu] (-18.00161479678271,-3.986522386252468) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-5.9871493613302365) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-7.987776336408002) circle (1.5pt);
\draw [fill=uuuuuu] (-18.001614796782714,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-15.99649768126352,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-15.988056217233657,-7.987776336408002) circle (1.5pt);
\draw [fill=uuuuuu] (-15.979614753203792,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-15.971173289173926,-3.9865223862524677) circle (1.5pt);
\draw [fill=uuuuuu] (-15.96278448206971,-1.9983751025525325) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-2.006906975602861) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-7.987776336408002) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-12.00393571799508,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-12.012522099862366,-7.987776336408001) circle (1.5pt);
\draw [fill=uuuuuu] (-12.021108481729653,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-12.02969486359694,-3.986522386252468) circle (1.5pt);
\draw [fill=uuuuuu] (-12.038156008390947,-2.015075649249038) circle (1.5pt);
\draw [fill=uuuuuu] (-10.002939531650116,-2.0237361448947433) circle (1.5pt);
\draw [fill=uuuuuu] (-10.01139981717321,-3.9865223862524677) circle (1.5pt);
\draw [fill=uuuuuu] (-10.020023209307166,-5.987149361330236) circle (1.5pt);
\draw [fill=uuuuuu] (-10.028646601441123,-7.987776336408001) circle (1.5pt);
\draw [fill=uuuuuu] (-10.037269993575077,-9.988403311485765) circle (1.5pt);
\draw [fill=uuuuuu] (-13.957794315242527,-3.986522386252468) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}

Motto: a finitely presented group $\langle S|R\rangle$ is a metric space. \pause

Caveat: up to quasiisometry (passing to an equivalent distance).\pause

Need to focus on rough, large scale features.\pause

Allows to use continuous tools.

\end{center}

\end{frame}

\section{Getting a feeling for GGT}

\subsection{Filling loops}

\begin{frame}

\textbf{The word problem}. Let $\langle S|R \rangle$ be a finite presentation of a group $G$. Find an algorithm that, given as input a (reduced) word $w$ in the alphabet $S$, returns YES if and only if $w$ is trivial in $G$, i.e. a product of conjugates of relators (elements of $R$).

\medskip

\textbf{Quantitative version}. Estimate the minimal number $A(w)$ of relators in terms of the length $|w|$ of $w$, i.e. the function 
$$
Dehn^{\langle S|R \rangle}:n\mapsto \max\{A(w)\,;\,|w|\leq n\}.
$$

\textbf{Examples}. In a free group, $A(w)=0$. In a free abelian group, $A(w)\sim \frac{1}{16}|w|^2$.\pause

\begin{theorem}[P. Novikov 1952, W. Boone 1959]
There exist finitely presented groups with unsolvable word problem.
\end{theorem}

\textbf{Geometric interpretation}. In the cell complex of the presentation, a word $w$ trivial in $G$ corresponds to an edge-loop, $A(w)$ is the area of disk filling it.

This helps when showing that the growth type of $Dehn^{\langle S|R \rangle}$ depends only on the group $G$, not on the choice of $S$.

\medskip

\textbf{Question}. Which functions arise as $Dehn^{\langle S|R \rangle}$ of some finite presentation of a group? 

\end{frame}

\begin{frame}

\begin{theorem}[Papasoglu, Bowditch]
$Dehn^{\langle S|R \rangle}$ subquadratic $\implies$ $Dehn^{\langle S|R \rangle}$ at most linear.
\end{theorem}

\begin{theorem}[Birget, Olshanski, Rips, Sapir 2002]
Every at least quartic function that can be simulated by a Turing machine is $Dehn^{\langle S|R \rangle}$ of some finite presentation of a group.
\end{theorem}

\textbf{Question}. Fill the gap between quadratic and quartic.

\end{frame}

\subsection{Hyperbolic groups}

\begin{frame}

For a finitely presented group, a linear filling inequality turns out to be equivalent to
% In 1986, Gromov extracted 
the following \emph{hyperbolicity} axiom, a property satisfied by Lobatchevsky's nonEuclidean geometry (\textbf{M Gromov 1986}).

\begin{minipage}{4.5cm}
\definecolor{ttttff}{rgb}{0.2,0.2,1.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.2cm,y=0.2cm]
\clip(-1.2867913836902547,-9.191692647375337) rectangle (13.119603291911934,6.464571885927483);
\draw [shift={(-2.698351795049233,0.9144192286454016)}] plot[domain=-1.1353215603123878:0.5299301814845291,variable=\t]({1.*7.010823369440908*cos(\t r)+0.*7.010823369440908*sin(\t r)},{0.*7.010823369440908*cos(\t r)+1.*7.010823369440908*sin(\t r)});
\draw [shift={(7.610665521290245,-18.935755740529512)},color=ttttff]  plot[domain=1.3540075674096879:2.0696505873486872,variable=\t]({1.*15.366350037843501*cos(\t r)+0.*15.366350037843501*sin(\t r)},{0.*15.366350037843501*cos(\t r)+1.*15.366350037843501*sin(\t r)});
\draw [shift={(25.474731668173796,16.80773544287817)}] plot[domain=3.6507096993401853:4.100270312731848,variable=\t]({1.*25.337238062361568*cos(\t r)+0.*25.337238062361568*sin(\t r)},{0.*25.337238062361568*cos(\t r)+1.*25.337238062361568*sin(\t r)});
\draw [shift={(-0.6829197675774682,0.7960420901547448)}] plot[domain=-1.5113201252168702:0.6362374137783968,variable=\t]({1.*8.654493715193993*cos(\t r)+0.*8.654493715193993*sin(\t r)},{0.*8.654493715193993*cos(\t r)+1.*8.654493715193993*sin(\t r)});
\draw [shift={(22.940818885699183,14.974277805564288)}] plot[domain=3.622349756074372:4.212277643365811,variable=\t]({1.*24.802485436870416*cos(\t r)+0.*24.802485436870416*sin(\t r)},{0.*24.802485436870416*cos(\t r)+1.*24.802485436870416*sin(\t r)});
\draw [->] (3.350883614620037,4.458201851197499) -- (1.1799091252134026,3.0736017641721975);
\draw [->] (0.259100282413175,-5.442083074273402) -- (0.7524699630844828,-7.941822789674693);
\draw (1.1800570196662847,5.8) node[anchor=north west] {$\delta$};
\draw (-1.5,-6.2) node[anchor=north west] {$\delta$};
\begin{scriptsize}
\draw [fill=ffqqqq] (0.259100282413175,-5.442083074273402) circle (2.5pt);
\draw[color=ffqqqq] (0.4893394667264537,-4) node {$A$};
\draw [fill=ffqqqq] (3.350883614620037,4.458201851197499) circle (2.5pt);
\draw[color=ffqqqq] (3.5811227989333165,5.8) node {$C$};
\draw [fill=ffqqqq] (10.915885384913425,-3.929082720214726) circle (2.5pt);
\draw[color=ffqqqq] (11.146124569226703,-2.5) node {$B$};
\end{scriptsize}
\end{tikzpicture}

\end{minipage}
%\hskip0.9cm
\begin{minipage}{5.5cm}
\begin{defi}
Let $\delta\in\R_+$. Say a finitely generated group $G$ is $\delta$-hyperbolic if geodesic triangles in $G$ are $\delta$-thin: each side is contained in the $\delta$-neighborhood of the two others.
\end{defi}
\end{minipage}\pause

\bigskip

Hyperbolic groups arise in topology: free groups, fundamental groups of higher genus surfaces are hyperbolic.\pause

\medskip

Hyperbolic groups are plentiful: 

\begin{itemize}
  \item Random groups (picking large random relators) are hyperbolic.
  \item Adding large relators does not kill the group.
  \item Taking limits of hyperbolic groups produces heaps of examples and counterexamples of weird finitely generated groups.
\end{itemize}
\end{frame}

\begin{frame}

\textbf{Geometry of hyperbolic groups}

\begin{minipage}{5.5cm}
Symmetry groups of nonEuclidean tilings are hyperbolic
\end{minipage}
\hskip1cm
\begin{minipage}{4cm}
\begin{center}
\includegraphics[width=1.4in]{kang3-9.pdf}

\vskip-.3cm
\hfill \tiny{G. Cousineau}
\end{center}
\end{minipage}\pause

\vskip.5cm

\begin{minipage}{5.5cm}
A hyperbolic group possesses an ideal boundary: a fractal space equipped with a \emph{quasisymmetric structure} allowing a new kind of real analysis: \emph{Analysis on Metric Spaces}. 

\smallskip

Keywords: differentiability, function spaces.
\end{minipage}
\hskip1cm
\begin{minipage}{4cm}
\begin{center}
\includegraphics[height=1.2in]{indra028.jpg}
\end{center}
\end{minipage}

\end{frame}

\section{Persistence}
\subsection{Persistence for the norm}

\begin{frame}

In Cayley graphs of groups, the distance $g\mapsto |g|$ to $e$ sometimes achieves local maxima.

\smallskip

\textbf{Example}. $G=\mathbb{Z}$ with generating system $\{2,3\}$. 

\definecolor{qqffqq}{rgb}{0.,1.,0.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-2.68,-1.16) rectangle (18.28,3.46);
\draw [shift={(1.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(3.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(5.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(7.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(9.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(11.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(2.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(4.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(6.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(8.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(10.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(12.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(13.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(14.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(15.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw (-2,1.4) node[anchor=north west] {-2};
\draw (0.02,1.4) node[anchor=north west] {-1};
\draw (2.04,1.4) node[anchor=north west] {0};
\draw (4.02,1.4) node[anchor=north west] {1};
\draw (5.96,1.4) node[anchor=north west] {2};
\draw (8.,1.4) node[anchor=north west] {3};
\draw (10.04,1.4) node[anchor=north west] {4};
\draw (11.98,1.4) node[anchor=north west] {5};
\draw (13.98,1.4) node[anchor=north west] {6};
\begin{scriptsize}
\draw [fill=qqqqff] (-1.,1.) circle (2.5pt);
\draw [fill=black] (3.,1.) circle (2.5pt);
\draw [fill=ffqqqq] (1.,1.) circle (2.5pt);
\draw [fill=ffqqqq] (5.,1.) circle (2.5pt);
\draw [fill=qqqqff] (7.,1.) circle (2.5pt);
\draw [fill=qqqqff] (9.,1.) circle (2.5pt);
\draw [fill=ffqqqq] (11.,1.) circle (2.5pt);
\draw [fill=ffqqqq] (13.,1.) circle (2.5pt);
\draw [fill=ffqqqq] (15.,1.) circle (2.5pt);
\draw [fill=qqffqq] (17.,1.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

Other view, showing distance to $0$:

\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\draw [color=cqcqcq,, xstep=0.5cm,ystep=0.5cm] (5.931422222222229,-7.7459111111111225) grid (20.4621777777778,-0.8085777777777785);
\clip(5.931422222222229,-7.7459111111111225) rectangle (20.4621777777778,-0.8085777777777785);
\draw (9.,-6.)-- (8.,-5.);
\draw (9.,-6.)-- (7.,-5.);
\draw (9.,-6.)-- (10.,-5.);
\draw (9.,-6.)-- (11.,-5.);
\draw (8.,-5.)-- (10.,-4.);
\draw (10.,-4.)-- (8.,-4.);
\draw (8.,-4.)-- (10.,-5.);
\draw (10.,-5.)-- (11.,-4.);
\draw (11.,-4.)-- (10.,-4.);
\draw (8.,-5.)-- (7.,-4.);
\draw (7.,-4.)-- (8.,-4.);
\draw (11.,-5.)-- (12.,-4.);
\draw (11.,-5.)-- (13.,-4.);
\draw (12.,-4.)-- (14.,-3.);
\draw (13.,-4.)-- (15.,-3.);
\draw (11.,-4.)-- (13.,-3.);
\draw (13.,-4.)-- (14.,-3.);
\draw (12.,-4.)-- (13.,-3.);
\draw [shift={(12.,-4.72)}] plot[domain=0.6240230529767566:2.5175696006130366,variable=\t]({1.*1.232233744059949*cos(\t r)+0.*1.232233744059949*sin(\t r)},{0.*1.232233744059949*cos(\t r)+1.*1.232233744059949*sin(\t r)});
\draw [shift={(14.,-3.70153846153846)}] plot[domain=0.6117577428097033:2.5298349107800897,variable=\t]({1.*1.2215384615384608*cos(\t r)+0.*1.2215384615384608*sin(\t r)},{0.*1.2215384615384608*cos(\t r)+1.*1.2215384615384608*sin(\t r)});
\draw (13.,-3.)-- (15.,-2.);
\draw (14.,-3.)-- (16.,-2.);
\draw (15.,-3.)-- (17.,-2.);
\draw (9.00751111111112,-6.261644444444454) node[anchor=north west] {0};
\draw (10.104577777777788,-5.15382222222223) node[anchor=north west] {2};
\draw (11.244666666666678,-5.143066666666674) node[anchor=north west] {3};
\draw (7.7276,-5.143066666666674) node[anchor=north west] {-2};
\draw (6.652044444444451,-5.164577777777786) node[anchor=north west] {-3};
\draw (7.749111111111119,-4.218088888888895) node[anchor=north west] {-1};
\draw (9.878711111111121,-4.185822222222228) node[anchor=north west] {1};
\draw (10.943511111111123,-4.228844444444451) node[anchor=north west] {4};
\draw (12.072844444444456,-4.164311111111117) node[anchor=north west] {5};
\draw (13.212933333333346,-4.175066666666673) node[anchor=north west] {6};
\draw (13.030088888888903,-3.196311111111116) node[anchor=north west] {7};
\draw (14.094888888888903,-3.2070666666666714) node[anchor=north west] {8};
\draw (15.138177777777793,-3.2070666666666714) node[anchor=north west] {9};
\draw (6.716577777777784,-4.196577777777784) node[anchor=north west] {-4};
\draw (14.,-3.)-- (15.,-2.);
\draw (15.,-3.)-- (16.,-2.);
\begin{scriptsize}
\draw [fill=qqqqff] (9.,-6.) circle (2.5pt);
\draw [fill=qqqqff] (8.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (7.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (10.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (11.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (10.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (8.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (11.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (7.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (12.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (13.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (14.,-3.) circle (2.5pt);
\draw [fill=qqqqff] (15.,-3.) circle (2.5pt);
\draw [fill=qqqqff] (13.,-3.) circle (2.5pt);
\draw [fill=qqqqff] (15.,-2.) circle (2.5pt);
\draw [fill=qqqqff] (16.,-2.) circle (2.5pt);
\draw [fill=qqqqff] (17.,-2.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

\end{frame}

\begin{frame}

In Cayley graphs of groups, the distance $g\mapsto |g|$ to $e$ sometimes achieves local maxima.

\smallskip

\textbf{Example}. $G=\mathbb{Z}$ with generating system $\{2,3\}$. 

\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\draw [color=cqcqcq,, xstep=0.5cm,ystep=0.5cm] (5.931422222222229,-7.7459111111111225) grid (20.4621777777778,-0.8085777777777785);
\clip(5.931422222222229,-7.7459111111111225) rectangle (20.4621777777778,-0.8085777777777785);
\draw (9.,-6.)-- (8.,-5.);
\draw (9.,-6.)-- (7.,-5.);
\draw (9.,-6.)-- (10.,-5.);
\draw (9.,-6.)-- (11.,-5.);
\draw (8.,-5.)-- (10.,-4.);
\draw (10.,-4.)-- (8.,-4.);
\draw (8.,-4.)-- (10.,-5.);
\draw (10.,-5.)-- (11.,-4.);
\draw (11.,-4.)-- (10.,-4.);
\draw (8.,-5.)-- (7.,-4.);
\draw (7.,-4.)-- (8.,-4.);
\draw (11.,-5.)-- (12.,-4.);
\draw (11.,-5.)-- (13.,-4.);
\draw (12.,-4.)-- (14.,-3.);
\draw (13.,-4.)-- (15.,-3.);
\draw (11.,-4.)-- (13.,-3.);
\draw (13.,-4.)-- (14.,-3.);
\draw (12.,-4.)-- (13.,-3.);
\draw [shift={(12.,-4.72)}] plot[domain=0.6240230529767566:2.5175696006130366,variable=\t]({1.*1.232233744059949*cos(\t r)+0.*1.232233744059949*sin(\t r)},{0.*1.232233744059949*cos(\t r)+1.*1.232233744059949*sin(\t r)});
\draw [shift={(14.,-3.70153846153846)}] plot[domain=0.6117577428097033:2.5298349107800897,variable=\t]({1.*1.2215384615384608*cos(\t r)+0.*1.2215384615384608*sin(\t r)},{0.*1.2215384615384608*cos(\t r)+1.*1.2215384615384608*sin(\t r)});
\draw (13.,-3.)-- (15.,-2.);
\draw (14.,-3.)-- (16.,-2.);
\draw (15.,-3.)-- (17.,-2.);
\draw (9.00751111111112,-6.261644444444454) node[anchor=north west] {0};
\draw (10.104577777777788,-5.15382222222223) node[anchor=north west] {2};
\draw (11.244666666666678,-5.143066666666674) node[anchor=north west] {3};
\draw (7.7276,-5.143066666666674) node[anchor=north west] {-2};
\draw (6.652044444444451,-5.164577777777786) node[anchor=north west] {-3};
\draw (7.749111111111119,-4.218088888888895) node[anchor=north west] {-1};
\draw (9.878711111111121,-4.185822222222228) node[anchor=north west] {1};
\draw (10.943511111111123,-4.228844444444451) node[anchor=north west] {4};
\draw (12.072844444444456,-4.164311111111117) node[anchor=north west] {5};
\draw (13.212933333333346,-4.175066666666673) node[anchor=north west] {6};
\draw (13.030088888888903,-3.196311111111116) node[anchor=north west] {7};
\draw (14.094888888888903,-3.2070666666666714) node[anchor=north west] {8};
\draw (15.138177777777793,-3.2070666666666714) node[anchor=north west] {9};
\draw (6.716577777777784,-4.196577777777784) node[anchor=north west] {-4};
\draw (14.,-3.)-- (15.,-2.);
\draw (15.,-3.)-- (16.,-2.);
\begin{scriptsize}
\draw [fill=qqqqff] (9.,-6.) circle (2.5pt);
\draw [fill=qqqqff] (8.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (7.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (10.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (11.,-5.) circle (2.5pt);
\draw [fill=qqqqff] (10.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (8.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (11.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (7.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (12.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (13.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (14.,-3.) circle (2.5pt);
\draw [fill=qqqqff] (15.,-3.) circle (2.5pt);
\draw [fill=qqqqff] (13.,-3.) circle (2.5pt);
\draw [fill=qqqqff] (15.,-2.) circle (2.5pt);
\draw [fill=qqqqff] (16.,-2.) circle (2.5pt);
\draw [fill=qqqqff] (17.,-2.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

Define the deadend depth of an element $g$ of $G$ with respect to a generating set $S$ as the distance of $g$ to the complement of the ball $B(e,|g|)$. 

\medskip

%\textbf{Example (T. Riley and S. Cleary 2005)}. There exists a finitely presented group and a generating system with unbounded deadend depth. 

%\medskip

\textbf{Example (T. Riley and A. Warshall 2006)}. There exists a finitely presented group with unbounded deadend depth for some generating system, and bounded deadend depth for an other generating system.

\end{frame}

\begin{frame}

\textbf{Example}. The \emph{lamplighter group} is $\langle a,t|a^2,\,[a^{t^n},a],\,\forall n\in\mathbb{Z}\rangle$. 

\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{xdxdff}{rgb}{0.49019607843137253,0.49019607843137253,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
%\draw [color=cqcqcq,, xstep=0.25cm,ystep=0.25cm] (1.8211111111111111,3.722222222222222) grid (12.47,6.62);
\clip(1.8211111111111111,3.9) rectangle (12.47,6.8);
\draw (2.23,6.006666666666665)-- (8.763333333333337,6.024444444444443);
\draw (2.2833333333333337,4.495555555555555)-- (8.754444444444449,4.513333333333333);
\begin{scriptsize}
\draw (5.3,5.8111111) node[anchor=north west] {0};
\draw (5.8,5.81111111111111) node[anchor=north west] {1};
\draw (6.3,5.81111111111111) node[anchor=north west] {2};
\draw (6.8,5.81111111111) node[anchor=north west] {3};
\draw (4.67,5.81111111111111) node[anchor=north west] {-1};
\draw (4.19,5.811111) node[anchor=north west] {-2};
\draw (3.71,5.811111) node[anchor=north west] {-3};
\draw [->,color=ffqqqq] (8.,5.5) -- (7.999939136921565,6.022367181324955);
\draw [->,color=ffqqqq] (5.5,4.) -- (5.499987932640823,4.504392518740466);
\draw (8.994444444444449,6.1133333333333315) node[anchor=north west] {$|g|=14$};
\draw (9.012222222222226,4.611111111111111) node[anchor=north west] {$|g_4|=25$};
%\begin{scriptsize}
\draw [fill=xdxdff] (2.4999798603925374,6.0074013057425635) circle (1pt);
\draw [fill=xdxdff] (2.9999761582588125,6.008761839886417) circle (1pt);
\draw [fill=xdxdff] (3.4999724561250876,6.010122374030271) circle (1pt);
\draw [fill=xdxdff] (3.999968753991363,6.011482908174124) circle (1pt);
\draw [fill=xdxdff] (4.499965051857638,6.012843442317978) circle (1pt);
\draw [fill=xdxdff] (4.999961349723914,6.0142039764618325) circle (1pt);
\draw [fill=xdxdff] (5.499957647590189,6.015564510605686) circle (1pt);
\draw [fill=xdxdff] (5.999953945456465,6.01692504474954) circle (1pt);
\draw [fill=xdxdff] (6.49995024332274,6.018285578893393) circle (1pt);
\draw [fill=xdxdff] (6.999946541189015,6.019646113037248) circle (1pt);
\draw [fill=xdxdff] (7.49994283905529,6.0210066471811015) circle (1pt);
\draw [fill=xdxdff] (7.999939136921565,6.022367181324955) circle (1pt);
\draw [fill=xdxdff] (8.49993543478784,6.023727715468809) circle (1pt);
\draw [fill=xdxdff] (2.500010574662906,4.496150822702065) circle (1pt);
\draw [fill=xdxdff] (3.0000068009925593,4.4975244387084645) circle (1pt);
\draw [fill=xdxdff] (3.500003027322212,4.498898054714865) circle (1pt);
\draw [fill=xdxdff] (3.9999992536518647,4.500271670721265) circle (1pt);
\draw [fill=xdxdff] (4.499995479981518,4.5016452867276655) circle (1pt);
\draw [fill=xdxdff] (4.99999170631117,4.503018902734065) circle (1pt);
\draw [fill=xdxdff] (5.499987932640823,4.504392518740466) circle (1pt);
\draw [fill=xdxdff] (5.999984158970475,4.505766134746866) circle (1pt);
\draw [fill=xdxdff] (6.499980385300129,4.507139750753265) circle (1pt);
\draw [fill=xdxdff] (6.999976611629781,4.508513366759666) circle (1pt);
\draw [fill=xdxdff] (7.499972837959434,4.509886982766066) circle (1pt);
\draw [fill=xdxdff] (7.999969064289087,4.511260598772466) circle (1pt);
\draw [fill=xdxdff] (8.49996529061874,4.512634214778866) circle (1pt);
\draw [fill=ffqqqq] (4.,6.5) circle (2.5pt);
\draw [fill=ffqqqq] (5.,6.5) circle (2.5pt);
\draw [fill=ffqqqq] (6.5,6.5) circle (2.5pt);
\draw [fill=ffqqqq] (8.,5.5) circle (1pt);
\draw [fill=ffqqqq] (4.,5.) circle (2.5pt);
\draw [fill=ffqqqq] (4.5,5.) circle (2.5pt);
\draw [fill=ffqqqq] (5.,5.) circle (2.5pt);
\draw [fill=ffqqqq] (5.5,5.) circle (2.5pt);
\draw [fill=ffqqqq] (6.,5.) circle (2.5pt);
\draw [fill=ffqqqq] (6.5,5.) circle (2.5pt);
\draw [fill=ffqqqq] (7.,5.) circle (2.5pt);
\draw [fill=ffqqqq] (7.5,5.) circle (2.5pt);
\draw [fill=ffqqqq] (5.5,4.) circle (1pt);
\draw [fill=ffqqqq] (3.5,5.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}

The element $g_n$ satisfies $|g_n|=6n+1$. $\forall h\in\partial B(g_n,n)$, $|h|\leq |g_n|-n$.

Hence a superlevel set $H^0$ barcode:

\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm]
\clip(-3.76,-2.) rectangle (6.14,3.82);
\draw [line width=3.6pt] (-3.,-1.)-- (0.,-1.);
\draw [line width=3.6pt] (-2.,0.)-- (1.16,-0.02);
\draw [line width=3.6pt] (-1.,1.)-- (2.3,1.);
\draw [line width=3.6pt] (0.,2.)-- (3.36,1.98);
\draw [line width=3.6pt] (1.,3.)-- (4.6,3.);
\end{tikzpicture}
\end{center}

Say a graph has \emph{very deep deadends} if there exists a vertex such that the distance barcode has arbitrarily long bars $(p,q)$ with $p=o(q)$.

\textbf{Question}. Do there exist finitely generated/presented groups with very deep deadends?

This is a quasiisometry invariant property.

\end{frame}

\subsection{Persistence for the length of loops}

\begin{frame}

On the \textbf{space of loops} in a Cayley graph, the length function sometimes achieves deep local minima, leading to $H^0$ barcodes with long bars.

\medskip

Define the \emph{loop shortening function} of a presentation $\langle S|R\rangle$ as follows: let $lsf:\mathbb{N}\to\mathbb{N}$ be the smallest function such that every loop of length $\leq n$ can be reduced to the constant loop by applications of relators one at a time via loops of length $\leq lsf(N)$.

\smallskip

This is a quasiisometry invariant.\pause

\medskip

\textbf{Easy fact}. For Novikov-Boone groups, $lsf$ cannot be bounded above by a recursive function.

\medskip

\textbf{Question}. Which functions arise as $lsf$ of some finite presentation of a group? \pause

\medskip

\textbf{Question}. What $lsf$ arise for specific subclasses of groups? 

\medskip

\textbf{Easy fact}. For hyperbolic groups, $lsf$ is linear.

\end{frame}

\section{Selfsimilarity}
\subsection{Nilpotent groups are asymptotically selfsimilar}

\begin{frame}

As a metric space, $\epsilon Cay(\mathbb{Z}^2)$ tends to $\R^2$ in $\ell^1$ metric as $\epsilon\to 0$. 

\smallskip

\begin{center}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.2cm,y=0.2cm]
\clip(-13.840079142916272,-8.829945737690847) rectangle (45.16869860428831,11.749865011615121);
\draw (-10.,10.)-- (-10.,-6.);
\draw (-6.,10.)-- (-6.,-6.);
\draw (-2.,10.)-- (-2.,-6.);
\draw (2.,10.)-- (2.,-6.);
\draw (-12.,8.)-- (4.,8.);
\draw (-12.,4.)-- (4.,4.);
\draw (-12.,0.)-- (4.,0.);
\draw (-12.,-4.)-- (4.,-4.);
\draw (8.,8.)-- (24.,8.);
\draw (8.,6.)-- (24.,6.);
\draw (8.,4.)-- (24.,4.);
\draw (8.,2.)-- (24.,2.);
\draw (8.,0.)-- (24.,0.);
\draw (8.,-2.)-- (24.,-2.);
\draw (8.,-4.)-- (24.,-4.);
\draw (10.,10.)-- (10.,-6.);
\draw (12.,10.)-- (12.,-6.);
\draw (14.,10.)-- (14.,-6.);
\draw (16.,10.)-- (16.,-6.);
\draw (18.,10.)-- (18.,-6.);
\draw (20.,10.)-- (20.,-6.);
\draw (22.,10.)-- (22.,-6.);
\draw (27.,10.)-- (27.,-6.);
\draw (28.,10.)-- (28.,-6.);
\draw (29.,10.)-- (29.,-6.);
\draw (30.,10.)-- (30.,-6.);
\draw (31.,10.)-- (31.,-6.);
\draw (32.,10.)-- (32.,-6.);
\draw (33.,10.)-- (33.,-6.);
\draw (34.,10.)-- (34.,-6.);
\draw (35.,10.)-- (35.,-6.);
\draw (36.,10.)-- (36.,-6.);
\draw (37.,10.)-- (37.,-6.);
\draw (38.,10.)-- (38.,-6.);
\draw (39.,10.)-- (39.,-6.);
\draw (40.,10.)-- (40.,-6.);
\draw (26.,9.)-- (42.,9.);
\draw (26.,8.)-- (42.,8.);
\draw (26.,7.)-- (42.,7.);
\draw (26.,6.)-- (42.,6.);
\draw (26.,5.)-- (42.,5.);
\draw (26.,4.)-- (42.,4.);
\draw (26.,3.)-- (42.,3.);
\draw (26.,2.)-- (42.,2.);
\draw (26.,1.)-- (42.,1.);
\draw (26.,0.)-- (42.,0.);
\draw (26.,-1.)-- (42.,-1.);
\draw (26.,-2.)-- (42.,-2.);
\draw (26.,-3.)-- (42.,-3.);
\draw (26.,-4.)-- (42.,-4.);
\draw (26.,-5.)-- (42.,-5.);
\draw (41.,10.)-- (41.,-6.);
\begin{scriptsize}
\draw [fill=qqqqff] (-10.,8.) circle (2.5pt);
\draw [fill=qqqqff] (-6.,8.) circle (2.5pt);
\draw [fill=qqqqff] (-10.,4.) circle (2.5pt);
\draw [fill=qqqqff] (-6.,4.) circle (2.5pt);
\draw [fill=qqqqff] (-10.,0.) circle (2.5pt);
\draw [fill=qqqqff] (-6.,0.) circle (2.5pt);
\draw [fill=qqqqff] (-10.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (-6.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (-2.,8.) circle (2.5pt);
\draw [fill=qqqqff] (2.,8.) circle (2.5pt);
\draw [fill=qqqqff] (-2.,4.) circle (2.5pt);
\draw [fill=qqqqff] (2.,4.) circle (2.5pt);
\draw [fill=qqqqff] (-2.,0.) circle (2.5pt);
\draw [fill=qqqqff] (2.,0.) circle (2.5pt);
\draw [fill=qqqqff] (-2.,-4.) circle (2.5pt);
\draw [fill=qqqqff] (2.,-4.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}

\end{frame}

\begin{frame}

As a metric space, $\epsilon Cay(\mathbb{Z}^2)$ tends to $\R^2$ in $\ell^1$ metric as $\epsilon\to 0$. 

\smallskip

This extends to finitely generated nilpotent groups. Let $G$ be a nilpotent group generated by finite $S$, $X=Cay(G,S)$. Then 
$$
\lim_{\epsilon\to 0}\epsilon X=G',
$$
where $G'$ is a selfsimilar nilpotent Lie group ($G'$ admits a $1$-parameter group of automorphisms homothetic for a leftinvariant geodesic metric).\pause

\begin{minipage}{4.7cm}
\textbf{Example: Heisenberg group} 

\smallskip

$G=\{\begin{pmatrix}1&x&z\\0&1&y\\0&0&1\end{pmatrix}\,;\,x,y,z\in\mathbb{Z}\}$.

\smallskip

$\delta_t\begin{pmatrix}1&x&z\\0&1&y\\0&0&1\end{pmatrix}=\begin{pmatrix}1&tx&t^2 z\\0&1&ty\\0&0&1\end{pmatrix}$.
\end{minipage}\hskip.5cm
\begin{minipage}{4.5cm}
\includegraphics[width=2in]{std1.png}
\end{minipage}

\end{frame}

\begin{frame}

As a metric space, $\epsilon Cay(\mathbb{Z}^2)$ tends to $\R^2$ in $\ell^1$ metric as $\epsilon\to 0$. 

\smallskip

This extends to finitely generated nilpotent groups. Let $G$ be a nilpotent group generated by finite $S$, $X=Cay(G,S)$. Then 
$$
\lim_{\epsilon\to 0}\epsilon X=G'
$$
where $G'$ is a selfsimilar nilpotent Lie group ($G'$ admits a $1$-parameter group of automorphisms homothetic for a leftinvariant geodesic metric).

\bigskip

\textbf{Motto}: most large scale properties of $G$ and $G'$ coincide.\pause

\smallskip

\textbf{Example}: volume growth is the same for $G$ and for $G'$.

\smallskip

\textbf{$lsf$: question}. Is $lsf$ linear for all finitely generated nilpotent groups?\pause

\medskip

\textbf{$Dehn$: counterexample (C. Llosa-Isenrich, G. Pallier and R. Tessera 2019)}. There exists a finitely generated nilpotent group $G$ such that $Dehn^G(n)\sim n^3$ and $Dehn^{G'}(n)\sim n^4$.

\end{frame}

\subsection{$2$-parameter Dehn function}

\begin{frame}

$X$ metric space.
An $\eta$-filling of a loop $c:\partial D\to X$ is an extension of $c$ to the $1$-skeleton of a triangulation of the disk $D$ such that the images of the boundaries of triangles have length $\leq \eta$. Its area is the number of triangles.

\medskip

Define $Area_\eta(c)$ as the the minimum area of $\eta$-fillings of $c$. Define a function $Fill_\eta^X:\mathbb{N}\to\mathbb{N}$ by
$$
Fill_\eta^X(n)=\sup\{Area_\eta(c)\,;\,Length(c)\leq n\}.
$$
This is a quasiisometry invariant (up to changing $\eta$).\pause

\bigskip

\textbf{Proposition}. For Cayley polyhedra of finite presentations, $Fill_\eta \sim Dehn$ (with constants depending on $\eta$).\pause

\medskip

Since $Fill_\eta^{\epsilon X}(n)=Fill_{\eta/\epsilon}^{X}(n/\epsilon)$, understanding the discontinuity of $\epsilon\mapsto Dehn^{\epsilon G}$ is equivalent to understanding $Fill_\eta^{X}(n)$ as a $2$-variable function.

\medskip

Alternatively, $Fill_{\eta/\epsilon}^{X}(n/\epsilon)=Dehn^{S(\epsilon),R(\epsilon)}(n)$ where $S(\epsilon)$ is the ball of radius $1/\epsilon$ and $R(\epsilon)$ the set of trivial words of length $\eta/\epsilon$.

\end{frame}

\end{document}

\begin{frame}

\textbf{Growth}: Count number $v(n)$ of vertices in $n$-ball.

Invariant up to $n\to Cn$.
\pause

\bigskip

\begin{minipage}{5.5cm}
\begin{center}
Free abelian group

$\langle a,b|aba^{-1}b^{-1}\rangle$ 

$v(n)=2n^2+2n+1$.
\end{center}
\end{minipage}
\hskip-.91cm
\begin{minipage}{5cm}
\begin{center}
Free group

$\langle a,b|\rangle$ 

$v(n)=4.3^{n-1}$
\end{center}
\end{minipage}\pause

\begin{thm}[Misha Gromov 1982]
Finitely generated group $G$ has polynomial growth $\iff$ $G$ is virtually nilpotent.
\end{thm}

%\textbf{Corollary}. The class of finitely generated virtually nilpotent groups is \emph{rigid}: a group quasiisometric to a group in the class belongs to the class.\pause

\begin{thm}[Rostislav Grigorchuk 1984]
There are finitely generated groups with growth between $e^{n^\alpha}$ and $e^{n^\beta}$, $0<\alpha<\beta<1$.
\end{thm}\pause

\begin{thm}[Anna Erschler - Tianyi Zheng 2018]
The growth of Grigorchuk's example satisfies
$\frac{\log\log v(n)}{\log n}\to \alpha_0$, $\alpha_0=0.7674...$.
\end{thm}
Random walks are used as a tool.

\end{frame}

%\subsection{Question}

\begin{frame}

\textbf{Question}: What are the possible growths for finitely generated groups?

\begin{conj}[Grigorchuk]
If not polynomial, growth cannot be $<e^{\sqrt{n}}$.
\end{conj}\pause

\bigskip

Using Lie algebras, Laurent Bartholdi and Grigorchuk (2000) solved the case of residually $p$ groups.

\bigskip

Yehuda Shalom and Terry Tao's effective version of Gromov's theorem (2009) implies:
$$
v(n)\leq n^{c(\log\log n)^c}\implies G\text{ is virtually nilpotent}.
$$

\end{frame}




