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71-476: Bass–Serre theory was developed by Jean-Pierre Serre in the 1970s and formalized in Trees , Serre's 1977 monograph (developed in collaboration with Hyman Bass ) on the subject. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. However, the theory quickly became a standard tool of geometric group theory and geometric topology , particularly

142-433: A i are non-zero integers (but n may be zero). In less formal terms, the group consists of words in the generators and their inverses , subject only to canceling a generator with an adjacent occurrence of its inverse. If G is any group, and S is a generating subset of G , then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G . For example,

213-437: A finitely generated group has more than one end if and only if this group admits a nontrivial splitting over finite subgroups that is, if and only if the group admits a nontrivial action without inversions on a tree with finite edge stabilizers. An important general result of the theory states that if G is a group with Kazhdan's property (T) then G does not admit any nontrivial splitting, that is, that any action of G on

284-489: A consecutive copy of a relator. The group elements are the equivalence classes, and the group operation is concatenation. This point of view is particularly common in the field of combinatorial group theory . A presentation is said to be finitely generated if S is finite and finitely related if R is finite. If both are finite it is said to be a finite presentation . A group is finitely generated (respectively finitely related , finitely presented ) if it has

355-806: A finite étale map – are important. This acted as one important source of inspiration for Grothendieck to develop the étale topology and the corresponding theory of étale cohomology . These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique (SGA) 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne . From 1959 onward Serre's interests turned towards group theory , number theory , in particular Galois representations and modular forms . Amongst his most original contributions were: his " Conjecture II " (still open) on Galois cohomology; his use of group actions on trees (with Hyman Bass );

426-401: A finite presentation ⟨ S | R ⟩ is just | S | − | R | and the deficiency of a finitely presented group G , denoted def( G ), is the maximum of the deficiency over all presentations of G . The deficiency of a finite group is non-positive. The Schur multiplicator of a finite group G can be generated by −def( G ) generators, and G is efficient if this number

497-466: A finitely generated projective module over a polynomial ring is free . This question led to a great deal of activity in commutative algebra , and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976. This result is now known as the Quillen–Suslin theorem . Serre, at twenty-seven in 1954, was and still is the youngest person ever to have been awarded

568-445: A finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. From this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that there are uncountably many non-isomorphic two generator groups. Therefore, there are finitely generated groups that cannot be recursively presented. One of

639-460: A free product decomposition where F ( X ) is a free group with free basis X = E ( A − T ) consisting of all positively oriented edges (with respect to some orientation on A ) in the complement of some spanning tree T of A . Let g be an element of G = π 1 ( A , T ) represented as a product of the form where e 1 , ..., e n is a closed edge-path in A with the vertex sequence v 0 , v 1 , ..., v n = v 0 (that

710-462: A fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures . Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC, 1955), on coherent cohomology , and Géométrie Algébrique et Géométrie Analytique ( GAGA , 1956). Even at an early stage in his work Serre had perceived

781-507: A graph of groups consisting of a single loop-edge e (together with its formal inverse e ), a single vertex v = o ( e ) = t ( e ), a vertex group B  =  A v , an edge group C  =  A e and the boundary monomorphisms α = α e : C → B , ω = ω e : C → B {\displaystyle \alpha =\alpha _{e}:C\to B,\omega =\omega _{e}:C\to B} . Then T = v

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852-600: A group B as a quotient of the free product subject to the relations This presentation can be rewritten as which shows that B is an iterated amalgamated free product of the vertex groups A v . Then the group G = π 1 ( A , T ) has the presentation which shows that G = π 1 ( A , T ) is a multiple HNN extension of B with stable letters { e | e ∈ E + ( A − T ) } {\displaystyle \{e|e\in E^{+}(A-T)\}} . An isomorphism between

923-430: A group G and the fundamental group of a graph of groups is called a splitting of G . If the edge groups in the splitting come from a particular class of groups (e.g. finite, cyclic, abelian, etc.), the splitting is said to be a splitting over that class. Thus a splitting where all edge groups are finite is called a splitting over finite groups. Algebraically, a splitting of G with trivial edge groups corresponds to

994-496: A need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field could not capture as much topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by

1065-492: A presentation is now the most common, earlier writers used different variations on the same format. Such notations include the following: Let S be a set and let F S be the free group on S . Let R be a set of words on S , so R naturally gives a subset of F S {\displaystyle F_{S}} . To form a group with presentation ⟨ S ∣ R ⟩ {\displaystyle \langle S\mid R\rangle } , take

1136-405: A presentation that is finitely generated (respectively finitely related, a finite presentation). A group which has a finite presentation with a single relation is called a one-relator group . If S is indexed by a set I consisting of all the natural numbers N or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering ) f  : F S → N from

1207-440: A presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group. A closely related but different concept is that of an absolute presentation of a group . A free group on a set S is a group where each element can be uniquely described as a finite length product of the form: where the s i are elements of S, adjacent s i are distinct, and

1278-448: A tree X without edge-inversions (that is, so that for every edge e of X and every g in G we have ge ≠ e ), one can define the natural notion of a quotient graph of groups A . The underlying graph A of A is the quotient graph X/G . The vertex groups of A are isomorphic to vertex stabilizers in G of vertices of X and the edge groups of A are isomorphic to edge stabilizers in G of edges of X . Moreover, if X

1349-407: A tree X without edge-inversions has a global fixed vertex. Let G be a group acting on a tree X without edge-inversions. For every g ∈ G put Jean-Pierre Serre Jean-Pierre Serre ( French: [sɛʁ] ; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology , algebraic geometry and algebraic number theory . He was awarded

1420-453: A very young age he was an outstanding figure in the school of Henri Cartan , working on algebraic topology , several complex variables and then commutative algebra and algebraic geometry , where he introduced sheaf theory and homological algebra techniques. Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration . Together with Cartan, Serre established

1491-521: Is v 0 = o ( e 1 ), v n = t ( e n ) and v i = t ( e i ) = o ( e i +1 ) for 0 < i < n ) and where a i ∈ A v i {\displaystyle a_{i}\in A_{v_{i}}} for i = 0, ..., n . Suppose that g  = 1 in G . Then The normal forms theorem immediately implies that the canonical homomorphisms A v → π 1 ( A , T ) are injective, so that we can think of

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1562-420: Is a finite product x 1 r 1 x 1 ... x m r m x m of members of such conjugates. It follows that each element of N , when considered as a product in D 8 , will also evaluate to 1; and thus that N is a normal subgroup of F . Thus D 8 is isomorphic to the quotient group F / N . We then say that D 8 has presentation Here the set of generators is S = { r , f   }, and

1633-476: Is a spanning tree in A and the fundamental group π 1 ( A , T ) is isomorphic to the HNN extension with the base group B , stable letter e and the associated subgroups H  = α( C ), K  = ω( C ) in B . The composition ϕ = ω ∘ α − 1 : H → K {\displaystyle \phi =\omega \circ \alpha ^{-1}:H\to K}

1704-406: Is a tree that comes equipped with a natural group action of the fundamental group π 1 ( A , v ) without edge-inversions. Moreover, the quotient graph A ~ / π 1 ( A , v ) {\displaystyle {\tilde {\mathbf {A} }}/\pi _{1}(\mathbf {A} ,v)} is isomorphic to A . Similarly, if G is a group acting on

1775-442: Is also denoted by ω e {\displaystyle \omega _{e}} . There are two equivalent definitions of the notion of the fundamental group of a graph of groups: the first is a direct algebraic definition via an explicit group presentation (as a certain iterated application of amalgamated free products and HNN extensions ), and the second using the language of groupoids . The algebraic definition

1846-446: Is also known as the structure theorem . One of the immediate consequences is the classic Kurosh subgroup theorem describing the algebraic structure of subgroups of free products . Consider a graph of groups A consisting of a single non-loop edge e (together with its formal inverse e ) with two distinct end-vertices u = o ( e ) and v = t ( e ), vertex groups H = A u , K = A v , an edge group C = A e and

1917-630: Is an equivariant isomorphism between the tree X and the Bass–Serre covering tree A ~ {\displaystyle {\tilde {\mathbf {A} }}} . More precisely, there is a group isomorphism σ: G → π 1 ( A , v ) and a graph isomorphism j : X → A ~ {\displaystyle j:X\to {\tilde {\mathbf {A} }}} such that for every g in G , for every vertex x of X and for every edge e of X we have j ( gx ) = g j ( x ) and j ( ge ) = g j ( e ). This result

1988-519: Is an isomorphism and the above HNN-extension presentation of G can be rewritten as In this case the Bass–Serre tree X = A ~ {\displaystyle X={\tilde {\mathbf {A} }}} can be described as follows. The vertex set of X is the set of cosets VX = { gB  : g ∈ G }. Two vertices gB and fB are adjacent in X whenever there exists b in B such that either fB  =  gbeB or fB = gbe B . The G -stabilizer of every vertex of X

2059-401: Is called the origin or the initial vertex of e and the vertex o ( e ) is called the terminus of e and is denoted t ( e ). Both loop-edges (that is, edges e such that o ( e ) =  t ( e )) and multiple edges are allowed. An orientation on A is a partition of E into the union of two disjoint subsets E and E so that for every edge e exactly one of the edges from

2130-428: Is closer to v than z we have [ A z  : ω e ( A e )] = 1, that is A z  = ω e ( A e ). An action of a group G on a tree X without edge-inversions is called trivial if there exists a vertex x of X that is fixed by G , that is such that Gx = x . It is known that an action of G on X is trivial if and only if the quotient graph of groups for that action

2201-423: Is conjugate to B in G and the stabilizer of every edge of X is conjugate to H in G . Every vertex of X has degree equal to [ B  :  H ] + [ B  :  K ]. Let A be a graph of groups with underlying graph A such that all the vertex and edge groups in A are trivial. Let v be a base-vertex in A . Then π 1 ( A , v ) is equal to the fundamental group π 1 ( A , v ) of

Bass–Serre theory - Misplaced Pages Continue

2272-401: Is defined using the formalism of groupoids . It turns out that for every choice of a base-vertex v and every spanning tree T in A the groups π 1 ( A , T ) and π 1 ( A , v ) are naturally isomorphic . The fundamental group of a graph of groups has a natural topological interpretation as well: it is the fundamental group of a graph of spaces whose vertex spaces and edge spaces have

2343-414: Is easier to state: First, choose a spanning tree T in A . The fundamental group of A with respect to T , denoted π 1 ( A , T ), is defined as the quotient of the free product where F ( E ) is a free group with free basis E , subject to the following relations: There is also a notion of the fundamental group of A with respect to a base-vertex v in V , denoted π 1 ( A , v ), which

2414-470: Is equal to gKg and the G -stabilizer of every vertex of X of type gH is equal to gHg . For an edge [ gH , ghK ] of X its G -stabilizer is equal to gh α( C ) h g . For every c  ∈  C and h  ∈ ' k  ∈  K' the edges [ gH , ghK ] and [ gH, gh α( c ) K ] are equal and the degree of the vertex gH in X is equal to the index [ H :α( C )]. Similarly, every vertex of type gK has degree [ K :ω( C )] in X . Let A be

2485-479: Is equal to its normal closure, so ⟨ G | K ⟩ = F G / K . Since the identity map is surjective, φ is also surjective, so by the First Isomorphism Theorem , ⟨ G | K ⟩ ≅ im( φ ) = G . This presentation may be highly inefficient if both G and K are much larger than necessary. Corollary. Every finite group has a finite presentation. One may take the elements of the group for generators and

2556-472: Is exactly the standard action of π 1 ( A , v ) on A ~ {\displaystyle {\tilde {A}}} by deck transformations . A graph of groups A is called trivial if A = T is already a tree and there is some vertex v of A such that A v = π 1 ( A , A ). This is equivalent to the condition that A is a tree and that for every edge e = [ u ,  z ] of A (with o ( e ) = u , t ( e ) = z ) such that u

2627-464: Is isomorphic to ⟨ S ∣ R ⟩ {\displaystyle \langle S\mid R\rangle } . It is a common practice to write relators in the form x = y {\displaystyle x=y} where x and y are words on S . What this means is that y − 1 x ∈ R {\displaystyle y^{-1}x\in R} . This has

2698-430: Is not necessarily the most efficient one possible. The last set of relations can be transformed into using σ i 2 = 1 {\displaystyle \sigma _{i}^{2}=1} . relations: An example of a finitely generated group that is not finitely presented is the wreath product Z ≀ Z {\displaystyle \mathbf {Z} \wr \mathbf {Z} } of

2769-469: Is trivial. Typically, only nontrivial actions on trees are studied in Bass–Serre theory since trivial graphs of groups do not carry any interesting algebraic information, although trivial actions in the above sense (e. g. actions of groups by automorphisms on rooted trees) may also be interesting for other mathematical reasons. One of the classic and still important results of the theory is a theorem of Stallings about ends of groups. The theorem states that

2840-582: The Cayley table for relations. The negative solution to the word problem for groups states that there is a finite presentation ⟨ S | R ⟩ for which there is no algorithm which, given two words u , v , decides whether u and v describe the same element in the group. This was shown by Pyotr Novikov in 1955 and a different proof was obtained by William Boone in 1958. Suppose G has presentation ⟨ S | R ⟩ and H has presentation ⟨ T | Q ⟩ with S and T being disjoint. Then The deficiency of

2911-806: The Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Born in Bages , Pyrénées-Orientales , to pharmacist parents, Serre was educated at the Lycée de Nîmes. Then he studied at the École Normale Supérieure in Paris from 1945 to 1948. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris . In 1956 he

Bass–Serre theory - Misplaced Pages Continue

2982-790: The Fields Medal . He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, and was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre (Centre National de la Recherche Scientifique, CNRS). He is a foreign member of several scientific Academies (US, Norway, Sweden, Russia,

3053-453: The dihedral group D 8 of order sixteen can be generated by a rotation, r , of order 8; and a flip, f , of order 2; and certainly any element of D 8 is a product of r ' s and f ' s. However, we have, for example, rfr = f , r = r , etc., so such products are not unique in D 8 . Each such product equivalence can be expressed as an equality to the identity, such as Informally, we can consider these products on

3124-441: The normal subgroup generated by the relations R . As a simple example, the cyclic group of order n has the presentation where 1 is the group identity. This may be written equivalently as thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called relators , distinguishing them from the relations that do include an equals sign. Every group has

3195-399: The quotient group The elements of S are called the generators of ⟨ S ∣ R ⟩ {\displaystyle \langle S\mid R\rangle } and the elements of R are called the relators . A group G is said to have the presentation ⟨ S ∣ R ⟩ {\displaystyle \langle S\mid R\rangle } if G

3266-454: The Bass–Serre tree X = A ~ {\displaystyle X={\tilde {\mathbf {A} }}} can be described as follows. The vertex set of X is the set of cosets Two vertices gK and fH are adjacent in X whenever there exists k  ∈  K such that fH = gkH (or, equivalently, whenever there is h  ∈  H such that gK = fhK ). The G -stabilizer of every vertex of X of type gK

3337-550: The Borel–Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form ; and the Serre conjecture (now a theorem) on mod- p representations that made Fermat's Last Theorem a connected part of mainstream arithmetic geometry . In his paper FAC, Serre asked whether

3408-498: The Legion of Honour (Grand Croix de la Légion d'Honneur) and Grand Cross of the Legion of Merit (Grand Croix de l'Ordre National du Mérite). A list of corrections , and updating, of these books can be found on his home page at Collège de France. Group presentation In mathematics , a presentation is one method of specifying a group . A presentation of a group G comprises a set S of generators —so that every element of

3479-670: The Royal Society, Royal Netherlands Academy of Arts and Sciences (1978), American Academy of Arts and Sciences , National Academy of Sciences , the American Philosophical Society ) and has received many honorary degrees (from Cambridge, Oxford, Harvard, Oslo and others). In 2012 he became a fellow of the American Mathematical Society . Serre has been awarded the highest honors in France as Grand Cross of

3550-526: The alphabet S ∪ S − 1 {\displaystyle S\cup S^{-1}} . In this perspective, we declare two words to be equivalent if it is possible to get from one to the other by a sequence of moves, where each move consists of adding or removing a consecutive pair x x − 1 {\displaystyle xx^{-1}} or x − 1 x {\displaystyle x^{-1}x} for some x in S , or by adding or removing

3621-896: The books of Allen Hatcher , Gilbert Baumslag , Warren Dicks and Martin Dunwoody and Daniel E. Cohen. Serre's formalism of graphs is slightly different from the standard formalism from graph theory . Here a graph A consists of a vertex set V , an edge set E , an edge reversal map E → E ,   e ↦ e ¯ {\displaystyle E\to E,\ e\mapsto {\overline {e}}} such that e ≠ e and e ¯ ¯ = e {\displaystyle {\overline {\overline {e}}}=e} for every e in E , and an initial vertex map o : E → V {\displaystyle o\colon E\to V} . Thus in A every edge e comes equipped with its formal inverse e . The vertex o ( e )

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3692-416: The boundary monomorphisms α = α e : C → H , ω = ω e : C → K {\displaystyle \alpha =\alpha _{e}:C\to H,\omega =\omega _{e}:C\to K} . Then T = A is a spanning tree in A and the fundamental group π 1 ( A , T ) is isomorphic to the amalgamated free product In this case

3763-402: The corresponding group is recursively presented . This usage may seem odd, but it is possible to prove that if a group has a presentation with R recursively enumerable then it has another one with R recursive. Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of Graham Higman states that

3834-609: The earliest presentations of a group by generators and relations was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus – a presentation of the icosahedral group . The first systematic study was given by Walther von Dyck , student of Felix Klein , in the early 1880s, laying the foundations for combinatorial group theory . The following table lists some examples of presentations for commonly studied groups. Note that in each case there are many other presentations that are possible. The presentation listed

3905-420: The form g i g j g k − 1 {\displaystyle g_{i}g_{j}g_{k}^{-1}} , where g i g j = g k {\displaystyle g_{i}g_{j}=g_{k}} is an entry in the multiplication table. The definition of group presentation may alternatively be recast in terms of equivalence classes of words on

3976-404: The free group on S to the natural numbers, such that we can find algorithms that, given f ( w ), calculate w , and vice versa. We can then call a subset U of F S recursive (respectively recursively enumerable ) if f ( U ) is recursive (respectively recursively enumerable). If S is indexed as above and R recursively enumerable, then the presentation is a recursive presentation and

4047-425: The fundamental groups of the vertex groups and edge groups, respectively, and whose gluing maps induce the homomorphisms of the edge groups into the vertex groups. One can therefore take this as a third definition of the fundamental group of a graph of groups. The group G = π 1 ( A , T ) defined above admits an algebraic description in terms of iterated amalgamated free products and HNN extensions . First, form

4118-417: The group can be written as a product of powers of some of these generators—and a set R of relations among those generators. We then say G has presentation Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R . Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by

4189-402: The group of integers with itself. Theorem. Every group has a presentation. To see this, given a group G , consider the free group F G on G . By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F G , therefore

4260-496: The intuitive meaning that the images of x and y are supposed to be equal in the quotient group. Thus, for example, r in the list of relators is equivalent with r n = 1 {\displaystyle r^{n}=1} . For a finite group G , it is possible to build a presentation of G from the group multiplication table , as follows. Take S to be the set elements g i {\displaystyle g_{i}} of G and R to be all words of

4331-433: The left hand side as being elements of the free group F = ⟨ r , f ⟩ , and let R = ⟨ rfrf , r , f ‍ ⟩ . That is, we let R be the subgroup generated by the strings rfrf , r , f ‍ , each of which is also equivalent to 1 when considered as products in D 8 . If we then let N be the subgroup of F generated by all conjugates x Rx of R , then it follows by definition that every element of N

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4402-456: The pair e , e belongs to E and the other belongs to E . A graph of groups A consists of the following data: For every e ∈ E {\displaystyle e\in E} the map α e ¯ : A e → A t ( e ) {\displaystyle \alpha _{\overline {e}}\colon A_{e}\to A_{t(e)}}

4473-413: The quotient of F S {\displaystyle F_{S}} by the smallest normal subgroup that contains each element of R . (This subgroup is called the normal closure N of R in F S {\displaystyle F_{S}} .) The group ⟨ S ∣ R ⟩ {\displaystyle \langle S\mid R\rangle } is then defined as

4544-418: The set of relations is R = { r = 1, f = 1, ( rf ) = 1} . We often see R abbreviated, giving the presentation An even shorter form drops the equality and identity signs, to list just the set of relators, which is { r , f , ( rf ) } . Doing this gives the presentation All three presentations are equivalent. Although the notation ⟨ S | R ⟩ used in this article for

4615-429: The study of 3-manifolds . Subsequent work of Bass contributed substantially to the formalization and development of basic tools of the theory and currently the term "Bass–Serre theory" is widely used to describe the subject. Mathematically, Bass–Serre theory builds on exploiting and generalizing the properties of two older group-theoretic constructions: free product with amalgamation and HNN extension . However, unlike

4686-473: The technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres , which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, and also made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus. In the 1950s and 1960s,

4757-441: The traditional algebraic study of these two constructions, Bass–Serre theory uses the geometric language of covering theory and fundamental groups . Graphs of groups , which are the basic objects of Bass–Serre theory, can be viewed as one-dimensional versions of orbifolds . Apart from Serre's book, the basic treatment of Bass–Serre theory is available in the article of Bass, the article of G. Peter Scott and C. T. C. Wall and

4828-475: The underlying graph A in the standard sense of algebraic topology and the Bass–Serre covering tree A ~ {\displaystyle {\tilde {\mathbf {A} }}} is equal to the standard universal covering space A ~ {\displaystyle {\tilde {A}}} of A . Moreover, the action of π 1 ( A , v ) on A ~ {\displaystyle {\tilde {\mathbf {A} }}}

4899-460: The vertex groups A v as subgroups of G . Higgins has given a nice version of the normal form using the fundamental groupoid of a graph of groups. This avoids choosing a base point or tree, and has been exploited by Moore. To every graph of groups A , with a specified choice of a base-vertex, one can associate a Bass–Serre covering tree A ~ {\displaystyle {\tilde {\mathbf {A} }}} , which

4970-629: Was elected professor at the Collège de France , a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil . The French mathematician Denis Serre is his nephew. He practices skiing, table tennis, and rock climbing (in Fontainebleau ). From

5041-404: Was the Bass–Serre covering tree of a graph of groups A and if G = π 1 ( A , v ) then the quotient graph of groups for the action of G on X can be chosen to be naturally isomorphic to A . Let G be a group acting on a tree X without inversions. Let A be the quotient graph of groups and let v be a base-vertex in A . Then G is isomorphic to the group π 1 ( A , v ) and there

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