Presentations and Free Groups

\(\DeclareMathOperator{\ann}{ann} \DeclareMathOperator{\Ass}{Ass} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\can}{can} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\cp}{CharPoly} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\cont}{cont} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\dim}{dim} \DeclareMathOperator{\edim}{embdim} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\eval}{eval} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Fun}{Fun} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\gcd}{gcd} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\height}{height} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\ker}{ker} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\max}{max} \DeclareMathOperator{\Min}{Min} \DeclareMathOperator{\mp}{MinPoly} \DeclareMathOperator{\mSpec}{mSpec} \DeclareMathOperator{\Nul}{Nul} \DeclareMathOperator{\ns}{nullspace} \DeclareMathOperator{\opp}{opp} \DeclareMathOperator{\Orb}{Orb} \DeclareMathOperator{\Out}{Out} \DeclareMathOperator{\pdim}{pdim} \DeclareMathOperator{\Perm}{Perm} \DeclareMathOperator{\PtStab}{pStab} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\sStab}{SetStab} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Span}{Span} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\trace}{trace} \DeclareMathOperator{\trdeg}{trdeg} \newcommand{\Ob}{\mathrm{Ob}} \newcommand{\Set}{\mathbf{Set}} \newcommand{\Grp}{\mathbf{Grp}} \newcommand{\Ab}{\mathbf{Ab}} \newcommand{\Sgrp}{\mathbf{Sgrp}} \newcommand{\Ring}{\mathbf{Ring}} \newcommand{\Fld}{\mathbf{Fld}} \newcommand{\cRing}{\mathbf{cRing}} \newcommand{\Mod}[1]{#1-\mathbf{Mod}} \newcommand{\Cx}[1]{#1-\mathbf{Comp}} \newcommand{\vs}[1]{#1-\mathbf{vect}} \newcommand{\Vs}[1]{#1-\mathbf{Vect}} \newcommand{\vsp}[1]{#1-\mathbf{vect}^+} \newcommand{\Top}{\mathbf{Top}} \newcommand{\Setp}{\mathbf{Set}_*} \newcommand{\Alg}[1]{#1-\mathbf{Alg}} \newcommand{\cAlg}[1]{#1-\mathbf{cAlg}} \newcommand{\PO}{\mathbf{PO}} \newcommand{\Cont}{\mathrm{Cont}} \newcommand{\MaT}[1]{\mathbf{Mat}_{#1}} \newcommand{\Rep}[2]{\mathbf{Rep}_{#1}(#2)} \def\l{\lambda} \def\a{\alpha} \def\b{\beta} \def\d{\delta} \def\e{\varepsilon} \def\g{\gamma} \def\t{\theta} \def\s{\sigma} \def\z{\zeta} \def\vp{\varphi} \newcommand{\A}{\mathbb{A}} \newcommand{\B}{\mathbb{B}} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\E}{\mathbb{E}} \newcommand{\F}{\mathbb{F}} \newcommand{\G}{\mathbb{G}} \newcommand{\H}{\mathbb{H}} \newcommand{\I}{\mathbb{I}} \newcommand{\J}{\mathbb{J}} \newcommand{\K}{\mathbb{K}} \newcommand{\L}{\mathbb{L}} \newcommand{\M}{\mathbb{M}} \newcommand{\N}{\mathbb{N}} \newcommand{\O}{\mathbb{O}} \newcommand{\P}{\mathbb{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\S}{\mathbb{S}} \newcommand{\T}{\mathbb{T}} \newcommand{\U}{\mathbb{U}} \newcommand{\V}{\mathbb{V}} \newcommand{\W}{\mathbb{W}} \newcommand{\X}{\mathbb{X}} \newcommand{\Y}{\mathbb{Y}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ON}{\mathbb{ON}} \def\cA{\mathcal A} \def\cB{\mathcal B} \def\cC{\mathcal C} \def\cD{\mathcal D} \def\cE{\mathcal E} \def\cF{\mathcal F} \def\cG{\mathcal G} \def\cH{\mathcal H} \def\cI{\mathcal I} \def\cJ{\mathcal J} \def\cK{\mathcal K} \def\cL{\mathcal L} \def\cM{\mathcal M} \def\cN{\mathcal N} \def\cO{\mathcal O} \def\cP{\mathcal P} \def\cQ{\mathcal Q} \def\cR{\mathcal R} \def\cS{\mathcal S} \def\cT{\mathcal T} \def\cU{\mathcal U} \def\cV{\mathcal V} \def\cW{\mathcal W} \def\cX{\mathcal X} \def\cY{\mathcal Y} \def\cZ{\mathcal Z} \newcommand{\fa}{{\mathfrak a}} \newcommand{\fb}{{\mathfrak b}} \newcommand{\fc}{{\mathfrak c}} \newcommand{\fd}{{\mathfrak d}} \newcommand{\fe}{{\mathfrak e}} \newcommand{\ff}{{\mathfrak f}} \newcommand{\fg}{{\mathfrak g}} \newcommand{\fh}{{\mathfrak h}} \newcommand{\fi}{{\mathfrak i}} \newcommand{\fj}{{\mathfrak j}} \newcommand{\fk}{{\mathfrak k}} \newcommand{\fl}{{\mathfrak l}} \newcommand{\fm}{{\mathfrak m}} \newcommand{\fn}{{\mathfrak n}} \newcommand{\fo}{{\mathfrak o}} \newcommand{\fp}{{\mathfrak p}} \newcommand{\fq}{{\mathfrak q}} \newcommand{\fr}{{\mathfrak r}} \newcommand{\fs}{{\mathfrak s}} \newcommand{\ft}{{\mathfrak t}} \newcommand{\fu}{{\mathfrak u}} \newcommand{\fv}{{\mathfrak v}} \newcommand{\fw}{{\mathfrak w}} \newcommand{\fx}{{\mathfrak x}} \newcommand{\fy}{{\mathfrak y}} \newcommand{\fz}{{\mathfrak z}} \newcommand{\fK}{{\mathfrak K}} \newcommand{\fR}{{\mathfrak R}} \def\sA{\mathscr A} \def\sB{\mathscr B} \def\sC{\mathscr C} \def\sD{\mathscr D} \def\sE{\mathscr E} \def\sF{\mathscr F} \def\sG{\mathscr G} \def\sH{\mathscr H} \def\sI{\mathscr I} \def\sJ{\mathscr J} \def\sK{\mathscr K} \def\sL{\mathscr L} \def\sM{\mathscr M} \def\sN{\mathscr N} \def\sO{\mathscr O} \def\sP{\mathscr P} \def\sQ{\mathscr Q} \def\sR{\mathscr R} \def\sS{\mathscr S} \def\sT{\mathscr T} \def\sU{\mathscr U} \def\sV{\mathscr V} \def\sW{\mathscr W} \def\sX{\mathscr X} \def\sY{\mathscr Y} \def\sZ{\mathscr Z} \def\adots{a_1, \dots, a_n} \def\bdots{b_1, \dots, b_n} \def\ddots{d_1, \dots, d_n} \def\edots{e_1, \dots, e_n} \def\fdots{f_1, \dots, f_n} \def\gdots{g_1, \dots, g_n} \def\udots{u_1, \dots, u_n} \def\xdots{x_1, \dots, x_n} \def\tS{\tilde{S}} \def\sdp{\rtimes} \newcommand{\tensor}{\otimes} \newcommand{\igen}[1]{\langle #1 \rangle} \def\nsg{\unlhd} \def\kval{{k-\mathrm{valued}}} \def\kalg{{k-\mathrm{alg}}} \newcommand\GG[2]{\Gal(#1/#2)} \newcommand{\MF}[3]{\Mat_{#1\times #2}(#3)} \newcommand{\vectwo}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}} \newcommand{\vecthree}[3]{\begin{bmatrix} #1 \\ #2 \\ #3\end{bmatrix}} \def\ob{{\mathfrak{ob}} } \def\qed{\square} \def\sse{\subseteq} \def\ss{\subset} \def\ssne{\subsetneq} \def\sm{\setminus} \def\inv{^{-1}} \newcommand{\es}{\emptyset} \newcommand{\Zm}[1]{\Z/({#1})} \def\ov#1{\overline{#1}} \newcommand{\leg}[2]{\left(\frac{{#1}}{{#2}}\right)} \def\th{^{th}} \def\htpy{\simeq_{\mathrm{htpc}}} \def\textand{ \, \text{and} \, } \def\textor{ \, \text{or} \, } \def\textfor{ \, \text{for} \, } \def\textfa{ \, \text{for all} \, } \def\textst{ \, \text{such that} \, } \def\textin{ \, \text{in} \, } \def\fg{ \, \text{finitely generated} \, } \newcommand{\op}{\mathrm{op}} \newcommand{\xra}[1]{\xrightarrow{#1}} \newcommand{\xora}[1]{\xtwoheadrightarrow{#1}} \newcommand{\xira}[1]{\xhookrightarrow{#1}} \newcommand{\xla}[1]{\xleftarrow{#1}} \def\lra{\longrightarrow} \def\into{\hookrightarrow} \def\onto{\twoheadrightarrow} \newcommand{\vv}[1]{\mathbf{#1}} \newcommand{\lm}[2]{{#1}\,\l + {#2}\,\mu} \renewcommand{\v}{\vv{v}} \renewcommand{\u}{\vv{u}} \newcommand{\w}{\vv{w}} \newcommand{\x}{\vv{x}} \renewcommand{\k}{\vv{k}} \newcommand{\0}{\vv{0}} \newcommand{\1}{\vv{1}} \newcommand{\vecs}[2]{#1_1,#1_2,\dots,#1_{#2}} \newcommand{\us}[1][n]{\vecs{\u}{#1}} \newcommand{\vs}[1][n]{\vecs{\v}{#1}} \newcommand{\ws}[1][n]{\vecs{\w}{#1}} \newcommand{\vps}[1][n']{\vecs{\v'}{#1}} \newcommand{\ls}[1][n]{\vecs{\l}{#1}} \newcommand{\mus}[1][n]{\vecs{\mu}{#1}} \newcommand{\lps}[1][n]{\vecs{\l'}{#1}} \def\td{\tilde{\delta}} \def\oo{\overline{\omega}} \def\ctJ{\tilde{\mathcal J}} \def\tPhi{\tilde{\Phi}} \def\te{\tilde{e}} \def\M{\operatorname{M}} \newcommand{\homotopic}{\simeq} \newcommand{\homeq}{\cong} \newcommand{\iso}{\approx} \newcommand{\dual}{\vee} \DeclarePairedDelimiter{\abs}{|}{|} \newcommand{\bv}{{\bar{v}}} \newcommand{\bu}{{\bar{u}}} \newcommand{\bw}{{\bar{w}}} \newcommand{\by}{{\bar{y}}} \newcommand{\ba}{{\bar{a}}} \newcommand{\bb}{{\bar{b}}} \newcommand{\bx}{{\bar{x}}} \DeclarePairedDelimiterX\setof[2]{\{}{\}}{#1\,|\,#2} \newcommand{\vx}{\underline{x}} \renewcommand{mod}[1]{\text{(mod }{#1})} \newcommand{\Slv}[3]{\sum_{{#2}=1}^{{#3}} {#1}_{{#2}} \v_{{#2}}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Chapter 4 Presentations and Free Groups

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